The Proportionality Principle of Simplicial Volumehopf.math.purdue.edu/Strohm/diploma_main.pdfample, Gromov’s proof of the Mostow rigidity theorem is based on the fol-lowing proportionality

The Proportionality Principleof Simplicial Volume

Clara Strohm

Westfalische Wilhelms-Universitat MunsterFachbereich Mathematik und Informatik

Diplomarbeit

The Proportionality Principleof Simplicial Volume

Clara Strohm

September 2004

Betreuer : Prof. Dr. Wolfgang LuckZweitgutachter: PD Dr. Michael Joachim

Hiermit versichere ich, daß ich diese Arbeit selbstandig verfaßt und nur dieangegebenen Quellen und Hilfsmittel benutzt habe.

Munster, im September 2004

Introduction

Manifolds, the basic objects of geometry, form the playground for both topo-logical and smooth structures. Many challenges in modern mathematics areconcerned with the nature of this interaction between algebraic topology,differential topology and differential geometry. For example all incarnationsof the index theorem, the Gauß-Bonnet formula, and the Mostow rigiditytheorem fall into this category.

Theorem (Mostow Rigidity Theorem). Each homotopy equivalence betweenoriented closed connected hyperbolic manifolds of dimension at least 3 is homotopicto an isometry.

This theorem was first proved by Mostow [Mostow]. But there is also aproof by Gromov [Munkholm], for which he designed the simplicial volume(therefore also known as “Gromov norm”).

The simplicial volume is a homotopy invariant of oriented closed connectedmanifolds defined in terms of the singular chain complex with real coeffi-cients. This invariant consists of a nonnegative real number measuring theefficiency of representing the fundamental class using singular simplices: IfM is an oriented closed connected manifold, then the simplicial volume of M,denoted by kMk , is the infimum of all `1-norms of singular cycles with real

v

coefficients representing the fundamental class. Since the fundamental classcan be viewed as a generalised triangulation of the manifold, the simplicialvolume can also be interpreted as a measure for the complexity of a mani-fold.

But the simplicial volume also has another important interpretation: it isa homotopy invariant approximation of the Riemannian volume. For ex-ample, Gromov’s proof of the Mostow rigidity theorem is based on the fol-lowing proportionality principle for hyperbolic manifolds [Gromov2, Sec-tion 2.2], [Thurston, Theorem 6.2]:

Theorem. For n 2 N define vn 2 R>0 as the maximal volume of an ideal n-simplex in n-dimensional hyperbolic space. Then for each hyperbolic oriented closedconnected manifold M of dimension n,

kMk

vol(M)=

1vn

.

In particular, the simplicial volume does not always vanish. However, ingeneral the simplicial volume is rather hard to compute. In the most caseswhere the simplicial volume is known, it is zero. This, for example, is thecase if the fundamental group of the manifold is Abelian or, more generally,amenable.

One of the most complete (but hardly digestible) references about the simpli-cial volume is Gromov’s pioneering paper “Volume and Bounded Cohomol-ogy” [Gromov2]. More recent introductions are given in the textbooks onhyperbolic geometry by Ratcliffe and Benedetti, Petronio [Ratcliffe, x11.5],[Benedetti, Petronio, Section C.3].

Applications of the simplicial volume also include knot theory [Murakami],the existence of certain S1-operations [Yano], and the investigation of othervolumes, such as the Riemannian or the minimal volume [Gromov2]. Fur-thermore, Gromov conjectured the following relation between the simplicialvolume and L2-invariants [Gromov3, Section 8A] – thereby connecting twodistant fields of algebraic topology:

Conjecture. Let M be an aspherical closed orientable manifold whose simplicialvolume vanishes. Then for all p 2 N,

b(2)p

�eM ;N (�1(M))

�= 0.

A rather complete account of L2-invariants is given in the book L2-Invariants:Theory and Applications to Geometry and K-theory [Luck2]. The above conjec-

vi

ture is justified by the observation that L2-invariants and the simplicial vol-ume show a similar behaviour. We will discuss this conjecture briefly inSection 1.4. Among the most striking similarities is the proportionality prin-ciple:

Theorem (Proportionality Principle of L2-Invariants). Let U be a simply con-nected Riemannian manifold. Then for each p 2 N there are constants Bp 2 R>0and a constant T 2 R>0 satisfying: for any discrete group G acting freely andcocompactly on U by isometries and all p 2 N,

b(2)p

�U;N (G)

�= Bp � vol(G nU).

If U with this G-action is of determinant class, then

%(2)�U;N (G)�= T � vol(G nU).

In particular, if M and N are oriented closed connected Riemannian man-ifolds with isometrically isomorphic universal Riemannian coverings, thenfor all p 2 N,

b(2)p

�eM ;N (�1(M))

�

vol(M)=

b(2)p

�eN ;N (�1(N))

�

vol(N)

and%(2)� eM ;N (�1(M))

�

vol(M)=

%(2)�eN ;N (�1(N))�

vol(N).

Theorem (Proportionality Principle of Simplicial Volume). Let M and N

be oriented closed connected Riemannian manifolds with isometrically isomorphicuniversal Riemannian coverings. Then

kMk

vol(M)=

kNk

vol(N).

These proportionality principles are also examples for an interesting linkbetween topological and differential structures. A similar proportionalityprinciple is known for Chern numbers [Hirzebruch, Satz 3].

A proof in the L2-case is given in Luck’s book [Luck2, Theorem 3.183]. Forthe simplicial volume, sketches of (different) proofs were given by Gromov[Gromov2, Section 2.3] and Thurston [Thurston, page 6.9]. However, thereis no complete proof of the proportionality principle of simplicial volume in

vii

the literature. It is the aim of this diploma thesis to provide such a proof,based on Thurston’s approach.

This text is organised as follows:

In the first chapter the simplicial volume is introduced and some of its prop-erties are collected.

Chapter 2 is devoted to the study of bounded cohomology, a quite peculiarvariant of singular cohomology. Its application as a basic tool for the analysisof the simplicial volume was discovered by Gromov [Gromov2]. However,we will mostly refer to the more elegant approach due to Ivanov, based onclassical homological algebra [Ivanov1], [Monod].

Thurston’s proof of the proportionality principle relies on the computationof the simplicial volume by means of a new homology theory called mea-sure homology [Thurston, pages 6.6–6.9]. Instead of linear combinations ofsingular simplices, measures on the set of (smooth) singular simplices areconsidered as chains, thereby permitting averaging constructions such asThurston’s smearing. But Thurston’s exposition lacks a proof of the fact thatmeasure homology and singular homology give rise to the same simplicialvolume. We will define measure homology in Chapter 3 and list its basic al-gebraic properties as they can be found in the papers of Zastrow and Hansen[Zastrow], [Hansen], in a slightly different setting.

Chapter 4 constitutes the central contribution of this diploma thesis. It willbe shown that measure homology indeed can be used to compute the simpli-cial volume, i.e., that singular homology with real coefficients is isometricallyisomorphic to measure homology. The proof is based on techniques frombounded cohomology, as explained in Chapter 2. It will also be discussedwhy Bowen’s argument [Bowen] is not correct (Subsection 4.3.4).

The last chapter finally contains the proof of the proportionality principle– using Thurston’s smearing technique. This forces us to study isometrygroups of Riemannian manifolds. En passant, it will be proved that thecompact open topology and the C1-topology (the smooth counterpart of thecompact open topology) on the isometry groups coincide. Unfortunately, Iwas not able to show this without making use of the theory of standard Borelspaces. The chapter closes with a proof of the fact that the simplicial volumeof a flat orientable closed connected manifold vanishes. The proof is basedon the proportionality principle, instead of referring to Gromov’s sophisti-cated estimate of the simplicial volume via the minimal volume [Gromov2,page 220].

I tried to keep the prerequisites for the understanding of this diploma thesis

viii

as small as possible. So only a basic familiarity with singular (co)homology,covering theory, measure theory, and Riemannian geometry is required. Avariety of textbooks is available providing such a background: e.g., [Lee1],[Lee2], [Massey], [Bredon], [Elstrodt]. But the detailed exposition of the sub-ject comes at a price – the length of the text, for which I apologise.

The symbols Z,N,Q,R,C stand for the set of integers, the set of nonnegativeintegers, the set of rational numbers, the set of real numbers, and the set ofcomplex numbers respectively. The one point space is written as �. Forsimplicity, we always assume that manifolds have dimension at least 1. The(co)homology class represented by an element x is denoted by [x]. All sums“P” are implicitly finite, unless stated otherwise. All notation is collected inthe table of notation (between bibliography and index).

Acknowledgements

First of all I would like to thank Professor Wolfgang Luck for giving me theopportunity to write this diploma thesis, for the encouragement, and for thefreedom I enjoyed during this project.

Roman Sauer and Marco Schmidt infected me with their fascination for thesimplicial volume and bounded cohomology. I am very grateful for all theirsupport and guidance. Marco Varisco was always open for discussions onmathematical principles and notation, and for making XYZ-theory accessi-ble for pedestrians like me. Mille grazie!

Professor Volker Puppe and his concise topology lectures and seminars ledme through my two years at the University of Konstanz. I would like tothank him for all his support.

Furthermore, I would like to thank Stefanie Helker for studying Englishand Benedikt Plitt, Julia Weber (Wohlfuhlburo!), and Michael Weiermannfor hunting bugs and for their valuable suggestions.

My parents always supported me, even though I did not manage to becomeacquainted with philosophy – but at least I am no gewm trhtoc anymore.Dankeschon!

All my gratitude is devoted to Andres Loh, not only for his TEXnical eksper-tise, but also for his patience and all the things language is not powerfulenough to express – except for Pluschtiere: Wuiisch? Zonder pf!

ix

Contents

1 Simplicial Volume 1

1.1 Singular (co)homologyrefreshed 2

1.2 Definition of thesimplicial volume 4

1.3 A zoo of properties 71.4 Simplicial volume and

L2-invariants 171.5 Generalisations 21

2 Bounded Cohomology 25

2.1 Bounded cohomologyof spaces 26

2.2 Duality 29

2.3 Bounded cohomologyof groups 35

2.4 The mapping theorem 402.5 A special resolution 44

x

3 Measure Homology 49

3.1 Prelude 503.2 Definition of measure

homology 60

3.3 Basic properties ofmeasure homology 63

4 Measure Homology is Singular Homology 71

4.1 Smooth singularhomology 72

4.2 The algebraicisomorphism 76

4.3 . . . is isometric 80

4.4 Integrating measurehomology chains 95

5 The Proportionality Principle 99

5.1 Statement of theproportionalityprinciple 100

5.2 Isometry groups 1025.3 The Haar measure 1115.4 Smearing 114

5.5 Proof of theproportionalityprinciple 120

5.6 Applications of theproportionalityprinciple 121

xi

xii

1 Simplicial Volume

The simplicial volume of an oriented closed connected manifold is a homotopyinvariant – defined in terms of the singular chain complex – which measures theefficiency of representing the fundamental class by singular chains (with real coef-ficients). Its first appearance is in Gromov’s famous proof of the Mostow rigiditytheorem [Munkholm].

Despite of being a topological invariant the simplicial volume contains interestinginformation about the possible differential structures on a manifold – in form ofthe Riemannian volume. Examples for such relations are Gromov’s estimate of theminimal volume and the proportionality principle, which is the topic of this thesis.

Moreover, there seems to be a deep connection to L2-invariants [Gromov3, Sec-tion 8A]:

Conjecture. Let M be an aspherical closed orientable manifold whose simplicial volumevanishes. Then for all p 2 N,

b(2)p

�eM ;N (�1(M))

�= 0.

In this chapter the simplicial volume is introduced (Section 1.2) and some of itsproperties are collected (Section 1.3). We then give a short survey about indicationssupporting the above conjecture in Section 1.4. In the last section we briefly discusssome generalisations of the simplicial volume.

1

1 Simplicial Volume

1.1 � Singular (co)homology refreshed

Before defining the simplicial volume, we recall the basic definitions of sin-gular (co)homology and introduce the corresponding notation which willbe used throughout this thesis. The key objects of singular (co)homologyare simplices:

Definition (1.1). For k 2 N the set

�k :=

�

(x0, : : : , xk) 2 Rk+1�0

��

k

Xj=0

xj = 1�

is called the k-dimensional standard simplex. For j 2 f0, : : : , kg we let

@j : �k�1�! �k

(x0, : : : , xk�1 ) 7�! (x0, : : : , xj�1 , 0, xj, : : : , xk�1 )

be the inclusion of the j-th face of �k. �

To be precise, the face mappings @j also depend on k, but we omit k in thenotation since it can easily be inferred from the context.

The main idea of the singular theory is to examine all continuous maps fromthe standard simplices to a topological space to detect the “holes” of thisspace, i.e., to measure which “cycles” are no “boundaries”.

Definition (1.2). Let X be a topological space and k 2 N.

� The set Sk (X) of singular k-simplices is the set of all continuous func-tions �k �! X. If � 2 Sk (X) and j 2 f0, : : : , kg, then � Æ@j is called thej-th face of � .

� Let R be a commutative ring with unit. Then the set Ck (X, R) of singu-lar k-chains of X with coefficients in R is the free R-module generatedby Sk (X).

2

1.1 Singular (co)homology refreshed

� The boundary operator @ : Ck+1 (X, R) �! Ck (X, R) is the R-modulehomomorphism given on the generators � 2 Sk+1 (X) by

@(�) :=k+1

Xj=0

(�1)j �� Æ@j.

The elements of im @ are called – as geometry suggests – boundaries,and the elements of ker @ are called cycles.

� We define the set Ck (X, R) of singular k-cochains of X with coeffi-cients in R as the algebraic dual of Ck (X, R), i.e.,

Ck (X, R) := HomZ

�Ck (X,Z) , R

�= HomR

�Ck (X, R), R

�

� The boundary operator @ : Ck+1 (X, R) �! Ck (X, R) induces the co-boundary operator Æ : Ck (X, R) �! Ck+1 (X, R) as follows:

Æ : Ck (X, R)�! Ck+1 (X, R)

f 7�!�c 7! (�1)k+1 � f(@c)

�.

The elements of im Æ are called coboundaries and the elements of ker Æ

are called cocycles.

In the special case R = Z, we use the abbreviations Ck (X) := Ck (X,Z) andCk (X) := Ck (X,Z) . �

A lengthy calculation shows that in the situation of the above definition theequality @ Æ @ = 0 holds (i.e., the singular chains form a chain complex).Therefore Æ ÆÆ = 0, too. Hence the following definition makes sense:

Definition (1.3). Let X be a topological space, k 2 N, and let R be a commu-tative ring with unit.

� The R-module

Hk (X, R) :=ker

�@ : Ck (X, R) ! Ck�1 (X, R)

�

im�@ : Ck+1 (X, R) ! Ck (X, R)

�

is called k-th singular homology of X with coefficients in R.

� The R-module

Hk (X, R) :=ker

�Æ : Ck (X, R) ! Ck+1 (X, R)

�

im�Æ : Ck�1 (X, R) ! Ck (X, R)

�

is called k-th singular cohomology of X with coefficients in R.

3

1 Simplicial Volume

For the sake of brevity we write

Hk (X) := Hk (X,Z) ,

Hk (X) := Hk (X,Z) . �

For convenience, we introduce the following convention:

Definition (1.4). Let M be an R-module, B � M, and (ab)b2B � R. Whenwe write Pb2B ab� b, we implicitly assume that all but a finite number of thecoefficients (ab)b2B are 0 unless stated otherwise. Additionally, wheneverwe write something like

“Let P�2Sk(X) a�� 2 Ck (X, R).”

it is understood that all a� are elements of R. �

Definition (1.5). Let f : X �! Y be a continuous map and let R be a com-mutative ring with unit. The map f induces chain maps

Ck (f , R) : Ck (X, R)7�! Ck (Y , R)

X�2Sk(X)

a��7�! X�2Sk(X)

a�� f Æ�

and

Ck (f , R) : Ck (Y , R)7�! Ck (X, R)

g 7�! gÆCk (f , R) .

The corresponding maps on (co)homology are denoted by H�(f , R) andH�(f , R) respectively. This turns singular (co)homology into a functor. �

More details on singular (co)homology can be found in any textbook on al-gebraic topology.

1.2 � Definition of the simplicial volume

We will first introduce a norm on the singular chain complex with real coef-ficients, the `1-norm. This norm induces a seminorm on singular homology.

4

1.2 De�nition of the simplicial volume

The simplicial volume of an oriented closed connected manifold is then de-fined as the seminorm of the fundamental homology class.

Definition (1.6). Let X be a topological space, and k 2 N. For all singularchains a = P�2Sk(X) a�� 2 Ck (X,R) we define

kak1 := X�2Sk(X)

ja�j. �

Since the sum is finite, this norm is well-defined and turns the singular chaingroup Ck (X,R) into a normed real vector space.

Definition (1.7). Let X be a topological space, k 2 N and � 2 Hk (X,R).Then k�k1 induces a seminorm as follows:

k�k1 := inf�kak1

��a 2 Ck (X,R), @(a) = 0, [a] = �

. �

In general, this is not a norm since the set of boundaries does not need tobe closed in the `1-norm. Geometrically, this seminorm measures how effi-ciently a homology class can be represented by singular chains.

The fundamental homology class of an n-dimensional oriented closed con-nected manifold M is the generator of Hn (M) �= Zwhich is compatible withthe orientation of M [Massey, Theorem XIV 2.2].

Definition (1.8). Let M be an oriented closed connected n-dimensional man-ifold. We will denote the corresponding fundamental homology class of Mby [M] 2 Hn (M) . Moreover, we write [M]R for the image of [M] under thechange of coefficients homomorphism Hn (M,Z) �! Hn (M,R) . �

Definition (1.9). The simplicial volume of an oriented closed connectedmanifold M is defined as

kMk := [M]R

1. �

Since k�k1 is not affected by multiplication with �1, the simplicial volumeis independent of the chosen orientation.

A good geometric description of [M] can be given in terms of triangulations[Bredon, page 338f]: if M is triangulable and �1, : : : , �m : �n �! M is a tri-angulation of M, then Pm

j=1 �j represents [M] . Hence the simplicial volumemeasures in some sense the complexity of M.

Of course, it is also possible to use other seminorms on singular homology(to get different “simplicial volumes”) or to generalise the above definitionto a larger class of spaces. We will sketch such extensions in Section 1.5.

5

1 Simplicial Volume

The following lemma indicates why the simplicial volume is akin to a vol-ume (for the simplicial volume, the simplices are all given the weight 1 in-stead of the integral expression, and only the absolute values of the coeffi-cients are considered).

Lemma (1.10). Let M be an oriented closed connected Riemannian manifold of di-mension n. If P�2Sn(M) a�� 2 Cn (M,R) is a cycle representing [M]R consistingof smooth singular simplices, then

vol(M) = X�2Sn(M)

a��

Z

�

volM .

Here volM denotes the volume form of M and the integral is explained in Defini-tion (4.29).

Proof. Since M is smooth, there is a triangulation �1, : : : , �m : �n �! M

[Whitehead]. Then one shows that the lemma is true for the cycle Pmj=1 �j

(which is intuitively clear) and uses the fact that the integral only depends onthe represented homology class [Lee2, Exercise 16.2 and Theorem 16.10].

Stronger connections – such as the proportionality principle – between thesimplicial volume and other volumes are discussed in Subsection 1.3.2 andChapter 5.

As a first example we will give a direct calculation for kS1k. There are muchmore sophisticated techniques for computing kS1k, but the following proofemphasises geometric intuition.

Theorem (1.11). Let X be an arcwise connected topological space. Then k�k1 = 0holds for all � 2 H1 (X,R).

Proof. The universal coefficient theorem yields

H1 (X) R �= H1 (X,R).

In other words: if � 2 H1 (X,R), then there exist k 2 N, r1, : : : , rk 2 R, and�1, : : : , �k 2 H1 (X) such that � corresponds to

k

Xj=1

rj � �j.

6

1.3 A zoo of properties

By the Hurewicz theorem each of the �j can be represented by a continuousfunction �j : S1 �! X. For each j 2 f1, : : : , kg and each d 2 N�1 we write�

(d)j for the composition (“wrapping �j around the circle d times”)

�j Æ (z 7! zd) : S1 �! X.

Now the Hurewicz theorem tells us that [�(d)j ] = d� [�j] holds in H1 (X) (and

thus also in H1 (X,R)). In particular,

k�k1 �

k

Xj=1

rj ��

(d)j

d

1�

1d�

k

Xj=1

jrjj.

Since this is true for any d 2 N�1 , we get k�k1 = 0.

Corollary (1.12). In particular, kS1k = 0.

This corollary also implies that the seminorm k�k1 on H1�S1,R

�is not a

norm.

It is more difficult to show that there are manifolds with non-zero simplicialvolume. For example, the simplicial volume of hyperbolic manifolds doesnot vanish (see Theorem (1.23)).

1.3 � A zoo of properties

In the first subsection, some elementary properties, such as homotopy in-variance and multiplicativity of the simplicial volume are deduced from thedefinition. We then give a collection of more sophisticated properties, basedon the comparison with other volumes and on the theory of bounded coho-mology. This section ends with a short (and rather incomplete) discussionof realisability.

1.3.1 � Multiplicativity

One of the main characteristics of the simplicial volume is that it is multi-plicative with respect to the degree of maps, which enables us to computethe simplicial volume in some special cases.

7

1 Simplicial Volume

We start with a simple – but fundamental – observation. Gromov calls thedescribed property functoriality [Gromov4, 5.34].

Lemma (1.13). Let f : X �! Y be a continuous map of topological spaces, k 2 Nand � 2 Hk(X,R). Then we get for the induced seminorm

kHk (f ,R) (�)k1 � k�k1 .

Proof. Let P�2Sk(X) a�� 2 Ck (X,R) be a cycle representing �. Then the classHk (f ,R) (�) is represented by P�2Sk(X) a�� f Æ� . Therefore

kHk (f ,R) (�)k1 =

�

X�2Sk(X)

a�� f Æ�

�

1�

X

�2Sk(X)

a�� f Æ�

1

� X�2Sk(X)

ja�j =

X

�2Sk(X)

a��

1.

Taking the infimum over all cycles representing � yields the desired inequal-ity kHk (f ,R) (�)k1 � k�k1 .

Multiplicativity (with respect to the degree of maps) is now a direct conse-quence of this lemma.

Definition (1.14). Let M and N be oriented closed connected n-dimensionalmanifolds and let [M] 2 Hn (M) , [N] 2 Hn (N) be their fundamental homol-ogy classes. Then the degree of a continuous map f : M �! N is defined asthe integer deg(f) satisfying

Hn (f)�[M]

�= deg(f)� [N] . �

Remark (1.15). In the situation of the definition,

Hn (f ,R)�[M]R

�= deg(f)� [N]R

also holds (by the universal coefficient theorem). �

Lemma (1.16). Let f : M �! N be a map of oriented closed connected manifoldsof the same dimension. Then

kMk � jdeg(f)j � kNk .

Proof. Using the previous lemma and the above remark we conclude that

kMk = [M]R

1

� H�(f ,R) ([M]R)

1

= deg(f)� [N]R

1

= jdeg(f)j � kNk .

8


Corollary (1.17). Since homotopy equivalences between oriented closed connectedmanifolds are maps of degree 1 or �1, the simplicial volume is a homotopy invariant.

Corollary (1.18). If f : M �! M is a self-map of an oriented closed connectedmanifold with jdeg(f)j � 2, then

kMk = 0.

In particular, kSnk = 0 for all n 2 N�1 and kS1 � S1k = 0.

Corollary (1.19). Let p : M �! N be a d-sheeted covering of oriented closed con-nected manifolds. Then

kMk = d�kNk .

Proof. Let n := dim N . Since p is a local homeomorphism, it also followsthat dim M = n. Of course, we may assume that p is orientation preserving.Thus deg(p) is well-defined and [Dold, Proposition VIII.4.7]

deg(p) = d.

Hence, kMk � d � kNk . As second step, we have to prove the “�” direc-tion: For � 2 Sn (N), let L(�) be the set of singular simplices e� 2 Sn (M)with p Æ e� = � . Since p is a d-sheeted covering and �n is arcwise con-nected and simply connected, the set L(�) contains precisely d elements.If a = P�2Sn(N) a�� 2 Cn (N ,R) represents [N]R , then

ea := X�2Sn(N)

a�� Xe�2L(�)

e�

represents [M]R because @(ea) = 0,

Hn (p,R)�[ea]�=

�

X�2Sn(N)

a�� d��

�

= d� [N]R

= Hn (p,R)�[M]R

�,

and deg(p) = d 6= 0. Thus

kMk = [M]R

1 � keak1 � d� X�2Sn(N)

ja�j = d�kak1 ,

which implies kMk � d� [N]R

1 = d�kNk .

9

1 Simplicial Volume

Using this result we can apply covering theory to obtain an upper boundfor the simplicial volume of oriented closed connected surfaces [Gromov2,page 217], [Benedetti, Petronio, Proposition C.4.7]:

Theorem (1.20). Let M be an oriented closed connected surface of genus g 2 N�2 .Then

kMk � 4g � 4.

To prove this theorem we need the following (slightly weaker) inequality:

Lemma (1.21). Let g 2 N�2 and let Fg be the oriented closed connected surface ofgenus g. Then

Fg

� 4g � 2.

Proof. The surface Fg can be constructed from a regular 4g-gon by identify-ing certain edges [Massey, x7 of Chapter I]. This 4g-gon can obviously betriangulated by 4g � 2 simplices:

This triangulation induces a “triangulation” of Fg consisting of 4g � 2 sim-plices �1, : : : , �4g�2 . This is no triangulation of Fg in the strict sense, but itssecond barycentric subdivision is. Hence we get [Bredon, page 338f] that

P4g�2j=1 �j is a cycle representing

�Fg

�R

, which yields the inequality Fg

� 4g � 2.

One can now combine the above inequality with a beautiful covering theoryargument to prove the theorem.

Proof (of Theorem (1.20)). For each d 2 N�2 there is a d-sheeted covering

pd : Md �! M

10


with connected covering space: we have [Massey, Example IV 5.3]

�1(Fg)ab �= Z2g ,

where �1(Fg)ab is the abelianisation of �1(Fg). Thus there is a surjective

group homomorphism

�1(M) �= �1(Fg) �! �1(Fg)ab �= Z2g �! Z.

The inverse image under this homomorphism of a subgroup of Z of index d

is a subgroup of �1(M) of index d. Therefore M admits a d-sheeted cover-ing pd : Md �! M (the covering corresponding to this subgroup) [Massey,Corollary V 7.5 and Theorem V 10.2].

Hence Md is also an oriented closed connected surface. As one can see byusing a CW-decomposition (or by calculating the volume if the manifoldsare hyperbolic),

�(Md) = d��(M).

Thus we get for the genus

g(Md) =��(Md) + 2

2=

�d��(M) + 22

= d� g � d+ 1,

i.e., Md and Fd�g�d+1are homotopy equivalent [Massey, Theorem I 8.2]. Ap-plying Corollary (1.19) and Lemma (1.21) results in

kMk =1d�kMdk =

1d� Fg(Md)

�1d��4g(Md) � 2

�

= 4g � 4+2d

.

This inequality holds for all d 2 N�2 . Therefore it follows that

kMk � 4g � 4.

The estimate of Theorem (1.20) is in fact an equality (Example (1.24)).

Additionally, one can use Corollary (1.19) to prove a toy version of the pro-portionality principle (Theorem (5.1)):

11

1 Simplicial Volume

Lemma (1.22). Let M and N be two oriented closed connected Riemannian man-ifolds. Suppose there is an oriented closed connected Riemannian manifold U ad-mitting locally isometric covering maps U �! M and U �! N , both with finitelymany sheets. Then

kMk

vol(M)=

kNk

vol(N).

Proof. In this situation, both the (Riemannian) volume and the simplicialvolume are multiplicative with respect to these coverings (Corollary (1.19)and [Lee2, Exercise 14.5]).

A proof of the general case requires much more effort. It will be givenin Chapter 5, based on the results about measure homology in Chapters3 and 4.

1.3.2 � Other volumes

There is a great variety of connections between the simplicial volume andother volumes arising in differential geometry. We have already seen a dis-tant relation between the simplicial volume and the Riemannian volume inLemma (1.10). Gromov used the following fact about hyperbolic manifoldsin his proof of Mostow rigidity:

Theorem (1.23). For n 2 N define vn 2 R>0 as the maximal volume of an ideal n-simplex in n-dimensional hyperbolic space. Then for each hyperbolic oriented closedconnected manifold M of dimension n,

kMk

vol(M)=

1vn

.

Proof. The estimate kMkvol M � 1

vncan be shown by using the so-called straight-

ening of simplices [Thurston, Corollary 6.1.7], [Ratcliffe, Lemma 3 in x11.4].It is possible to use the smearing technique (which will be explained in Sec-tion 5.4) to prove the reverse inequality [Thurston, Theorem 6.2], [Gromov2,page 235]. But there are also more direct proofs [Ratcliffe, Theorem 11.4.3].

In particular, there are oriented closed connected manifolds whose simpli-cial volume is not zero since vn < 1 [Benedetti, Petronio, Lemma C.2.3].

12


Example (1.24). Let M be an oriented closed connected surface of genus g 2N�2 . In view of the homotopy invariance of the simplicial volume we mayassume that M is hyperbolic (since there is a hyperbolic model of the sur-face with genus g – a good description of the hyperbolic structure on sur-faces is given in the book of Benedetti and Petronio [Benedetti, Petronio,Section B.3]). Hence the Gauß-Bonnet formula yields

vol(M) = 2� ��(M) = 2� � (2g � 2).

Moreover, v2 = � [Benedetti, Petronio, Lemma C.2.3]. Thus Theorem (1.23)implies

kMk = 4g � 4.

In particular, the simplicial volume can grow arbitrarily large and it is nobordism invariant. �

Based on straightening, Inoue and Yano found the following generalisationof Theorem (1.23) [Inoue, Yano]:

Theorem (1.25). For each n 2 N there is a constant Cn 2 R>0 such that thefollowing property holds: Let Æ 2 R>0 and let M be an n-dimensional orientedclosed connected Riemannian manifold whose sectional curvature is bounded fromabove by �Æ. Then

kMk � Cn �Æn/2 � vol(M).

The proportionality principle can also be seen as a generalisation of Theo-rem (1.23). Chapters 2, 3 and 4 provide the tools necessary for the proof(based on Thurston’s ideas) of this theorem in Chapter 5.

Theorem (1.26) (Proportionality Principle of Simplicial Volume). Let M

and N be oriented closed connected Riemannian manifolds with isometrically iso-morphic universal Riemannian coverings. Then

kMk

vol(M)=

kNk

vol(N).

Gromov showed that the simplicial volume also can be seen as a topolog-ical obstruction for the existence of certain Riemannian metrics [Gromov2,Section 0.5]:

Definition (1.27). Let M be a smooth manifold. Then its minimal volume,minvol(M), is the infimum over all volumes volg(M), where g is a com-plete Riemannian metric on M such that the sectional curvature satisfiesj secg(M)j � 1. �

13

1 Simplicial Volume

Example (1.28). Using the Gauß-Bonnet formula, one can show that the min-imal volume of the oriented closed connected surface of genus g 2 N�2is 2� � j2 � 2gj [Gromov2, page 213].

Scaling Riemannian metrics yields minvol(M) = 0 for all flat manifolds.

A rather counterintuitive result is that minvol(S3) = 0, as was found out byBerger [Gromov2, page 215]. �

According to Gromov, the minimal volume can be estimated from below asfollows [Gromov2, Section 0.5]:

Theorem (1.29). For all n-dimensional oriented closed connected smooth mani-folds M, the simplicial volume is bounded from above by the minimal volume via

kMk � (n � 1)n � n!�minvol(M).

Corollary (1.30). In particular, if M is an oriented closed connected flat manifold,then kMk = 0.

We will prove this corollary as an application of the proportionality principlein Section 5.6, without making use of Theorem (1.29).

1.3.3 � A dual point of view

Gromov also discovered that bounded cohomology can be used to inves-tigate the simplicial volume [Gromov2, Chapter 1]. We will sketch this ap-proach in Chapter 2. One of the main results is that the simplicial volume canbe computed in terms of bounded cohomology of groups (Corollary (2.32)).This has the following rather striking consequences:

Theorem (1.31). 1. If M is an oriented closed connected manifold with amena-ble fundamental group, then kMk = 0.

2. If f : M �! N is a map of degree 1 between oriented closed connected mani-folds of the same dimension such that ker �1(f) is amenable, then

kMk = kNk .

The class of amenable groups contains, for example, all Abelian groups andall finite groups, but it does not contain the free product Z�Z. In particular,we see again that kSnk = 0 for all n 2 N>0 and that kS1 � S1k = 0. Moredetails will be given in Sections 2.3.3 and 2.4.3.

14


Other consequences of the duality between k�k1 and bounded cohomologyare (Theorem (2.14) and Theorem (2.35)):

Theorem (1.32). 1. For each n 2 N>0 there is a constant c(n) 2 R>0 such thatthe following holds: if M and N are two oriented closed connected manifolds,then

kNk � kMk � kN �Mk � c(dim N + dim M)�kNk � kMk .

2. If M and N are two oriented closed connected manifolds of the same dimen-sion n � 3, then

kM # Nk = kMk + kNk .

In particular, for all oriented closed connected manifolds M, the simplicialvolume of the product S1 � M is zero. This is generalised in the followingway [Yano]:

Theorem (1.33). Suppose M is an oriented closed connected manifold admitting anon-trivial S1-action. Then kMk = 0.

1.3.4 � Which simplicial volumes are possible?

In this paragraph we want to analyse which values occur as the simplicialvolume of some oriented closed connected manifold.

For each v 2 R>0 there is an oriented closed connected Riemannian mani-fold M such that vol(M) = v. A very clumsy estimate, based on the follow-ing theorem, shows that this cannot be true for the simplicial volume:

Theorem (1.34). There are only countably many homotopy types CW-complexesconsisting of finitely many cells.

Proof. We prove by induction (on k 2 N) that there are only countably manyhomotopy types of CW-complexes having at most k cells.

If k = 0, there is nothing to show. For the induction step let k 2 N, and letY be a CW-complex consisting of k+ 1 cells. Then there is a CW-complex X

with k cells and an attaching map f : Sn �! X for some n 2 N such that

Y �= X [f Dn+1.

Here the homotopy type of Y depends only on the homotopy class of f andthe homotopy type of X [Lundell, Weingram, Corollary IV 2.4].

15

1 Simplicial Volume

By induction, there are only countably many homotopy types of such X.Hence we have to count the number of homotopy classes of possible attach-ing maps. Let s 2 Sn and x 2 X be two base points. If x is in the same con-nected component as f(Sn), then f is homotopic to a pointed map Sn �! X

because fsg ,! Sn is a cofibration. But there are only countably many ho-motopy classes of pointed maps Sn �! X because �n(X, x) is countable[Lundell, Weingram, Theorem IV 6.1]. Moreover, the finite CW-complex X

can only have finitely many connected components, and there are, of course,only countably many possible dimensions n 2 N.

Hence there can be only countably many homotopy types of CW-complexeswith k+ 1 cells.

Corollary (1.35). The set�kMk

��M is an oriented closed connected manifold

� R�0

is countable. In particular, not every positive real number can be realised as simpli-cial volume of some oriented closed connected manifold.

Proof. Each closed (oriented, connected) topological manifold has the ho-motopy type of a finite CW-complex [Kirby, Siebenmann, page 744]. Bythe previous theorem, there are only countably many homotopy types ofclosed (oriented, connected) manifolds. Now Corollary (1.17) completes theproof.

This corollary shows that the simplicial volume has a much more subtle be-haviour than the Riemannian volume: the simplicial volume cannot be af-fected by “scaling.”

The corollary gives, of course, a very unsatisfying and rough answer. Itwould be interesting to know whether the simplicial volumes lie denselyin R>0 . It would be sufficient to know that there is a sequence (Mn)n2N oforiented closed connected manifolds (of dimension at least 3) with non-zerosimplicial volume such that

limn!1

kMnk = 0.

Then one could use Gromov’s formula for connected sums (Theorem (1.32))to see that the simplicial volumes lie dense in R>0 .

Example (1.24) shows at least that the simplicial volume can become arbi-trarily large.

16

1.4 Simplicial volume and L2-invariants

The question of realisability has already been posed by Gromov [Gromov2,page 222]. He suspects that the simplicial volumes might form a closed well-ordered (with the standard ordering on the real line) non-discrete subsetof R. This speculation is based on a result of Thurston on hyperbolic mani-folds of dimension 3 [Gromov1], [Thurston]. However, until today there donot seem to exist any detailed answers to all these questions.

1.4 � Simplicial volume and L2-invariants

As mentioned in the introduction, there seems to be a deep connection be-tween the simplicial volume and L2-invariants, such as L2-Betti numbersand L2-torsion. More concretely, Gromov formulated the following conjec-ture [Gromov3, Section 8A]:

Conjecture. Let M be an aspherical closed orientable manifold whose simplicialvolume vanishes. Then for all p 2 N,

b(2)p

�eM ;N (�1(M))

�= 0.

Luck extended this conjecture as follows [Luck1, Conjecture 3.2]:

Moreover, eM is of determinant class and %(2)� eM ;N (�1(M))�= 0.

Since aspherical spaces are the central objects of concern, we recall the defi-nition:

Definition (1.36). A space is called aspherical if its universal covering iscontractible. (For CW-complexes this is the same as to require that all higherhomotopy groups vanish). �

The L2-Betti numbers b(2)� and L2-torsion %(2) can be defined (in the setting

of von Neumann algebras) similarly to their classic pendants [Luck2, Chap-ter 6].

Until now there does not seem to exist a strategy to prove (or disprove) theabove conjecture. However, there are many facts witnessing the probabletruth of this conjecture. In the following paragraphs we will present someof this evidence. A more extensive list can be found in Luck’s book [Luck2,Chapter 14]. For simplicity we refer to the proofs in this book instead to theoriginal papers.

17

1 Simplicial Volume

1.4.1 � The zeroth L2-Betti number

A rather easy argument shows that the above conjecture holds for the zerothL2-Betti number:

Lemma (1.37). Let M be an aspherical connected closed manifold. Then the funda-mental group �1(M) is torsion free and hence infinite.

Proof. Assume that �1(M) contains a finite cyclic subgroup G. Since M isaspherical and of finite dimension n, the cellular chain complex Ccell

� ( eM ;Z)is a bounded projective resolution of the trivial ZG-module Z. This impliesthat the group homology

H�(G;Z) �= H�

�Ccell� ( eM ;Z)ZG Z

�

vanishes in all degrees greater than n.

On the other hand, it is well-known that the group homology of nontriv-ial finite cyclic groups does not have this property [Weibel, Theorem 6.2.3].Hence G must be trivial, i.e., �1(M) is torsion free.

Corollary (1.38). If M is an aspherical closed connected manifold, then [Luck2,Theorem 6.54(8)]

b(2)0�eM ;N (�1(M))

�=

1j�1(M)j

= 0.

Example (1.39). The condition that the manifold be aspherical in Gromov’sconjecture is necessary: The manifold S2 is closed and orientable, but notaspherical because �2(S

2) �= Z. By Corollary (1.18) we have kS2k = 0. Onthe other hand [Luck2, Theorem 6.54(8)],

b(2)0�eS2;N (�1(S

2))�=

1j�1(S2)j

= 1 6= 0. �

1.4.2 � Amenability

If M is on oriented closed connected (not necessarily aspherical) manifoldwith amenable fundamental group, then kMk = 0 (Corollary (2.32)). Thecorresponding result in the L2-world is:

Theorem (1.40). Let M be an oriented closed connected aspherical manifold withamenable fundamental group. Then all L2-Betti numbers of M are zero.

18

1.4 Simplicial volume and L2-invariants

Proof. By Lemma (1.37), the fundamental group of M must be infinite. Butthen it can be shown that all L2-Betti numbers of M must be zero [Luck2,Corollary 6.75].

1.4.3 � Proportionality

One of the most remarkable similarities between the simplicial volume andL2-invariants is the proportionality principle. It is valid for the simplicialvolume (Theorem (5.1)) and for L2-Betti numbers (and L2-torsion) [Luck2,Theorem 3.183]:

Theorem (1.41) (Proportionality Principle of L2-Invariants). Let U be a sim-ply connected Riemannian manifold. Then for each p 2 N there are constants Bp 2R>0 and a constant T 2 R>0 satisfying: for any discrete group G acting freely andcocompactly on U by isometries and all p 2 N,

b(2)p

�U;N (G)

�= Bp � vol(G nU).

If U with this G-action is of determinant class, then

%(2)�U;N (G)�= T � vol(G nU).

1.4.4 � Multiplicativity

As we have seen in Section 1.3, the simplicial volume is multiplicative withrespect to the mapping degree. Therefore Gromov’s conjecture predicts thatif M is an aspherical oriented closed connected manifold admitting a self-map of degree at least two, then all L2-Betti numbers of M vanish. Thisstatement is known to be true if this self-map is a covering [Luck2, Exam-ple 1.37].

The general case is still open, but there is at least a partial result:

Definition (1.42). A group G is said to be Hopfian if every epimorphismG �! G is an isomorphism. �

Examples for Hopfian groups are finite groups, the integers, and, more gen-erally, residually finite groups. The group `1 (N,R) is not Hopfian as can beseen by considering the shift operator.

19

1 Simplicial Volume

Theorem (1.43). Let M be an aspherical oriented closed connected manifold. Ifthere is a continuous map f : M �! M such that jdeg(f)j � 2 and if all sub-groups of �1(M) of finite index are Hopfian, then

8p2N b(2)p

�eM ;N (�1(M))

�= 0.

A proof is given in [Luck2, Theorem 14.40]. (There only all normal subgroupsof �1(M) are required to be Hopfian – but this is a misprint.)

1.4.5 � Non-trivial S1-operations

Furthermore, the simplicial volume and the L2-Betti numbers behave simi-larly under S1-operations [Luck2, Corollary 1.43]:

Theorem (1.44). Let M be an aspherical closed smooth manifold with non-trivialS1-action. Then

8p2N b(2)p

�eM ;N (�1(M))

�= 0.

The corresponding statement about the simplicial volume is Theorem (1.33).

1.4.6 � Gromov’sconjecture

Looking at the conjecture and the known evidence one could also suspectthat the converse is true: If M is an aspherical oriented closed connectedmanifold whose L2-Betti numbers vanish, then kMk = 0.

But this statement is false: If M is an oriented closed connected hyperbolicmanifold of odd dimension, then

b(2)p

�eM ;N (�1(M))

�= 0

for all p 2 N [Luck2, Theorem 1.54]. On the other hand, kMk 6= 0 by Theo-rem (1.23).

20

1.5 Generalisations

1.5 � Generalisations

In the following, some possible generalisations of the simplicial volume aredescribed. In the first part, we consider the use of coefficients other than R.Later we sketch extensions of the simplicial volume to non-compact, non-connected, or non-orientable manifolds possibly with non-empty boundary,and also to Poincare complexes. In the last section we consider functorialseminorms in the sense of Gromov.

1.5.1 � Unreal coefficients

Instead of real coefficients we could also use the singular chain complexwith integral, complex or rational coefficients to define “simplicial volumes”kMkZ , kMkC, and kMkQ .

Definition (1.45). Let R 2 fZ,C,Qg and let M be an oriented closed con-nected manifold of dimension n. We write [M]R 2 Hn (M, R) �= R for theimage of the fundamental class [M] under the change of coefficients homo-morphism.

As in the case of singular homology with real coefficients, we can define an“R-linear norm” on Cn (M, R) via

X�2Sn(M)

a�� 7�! X�2Sn(M)

ja�j,

which induces a “seminorm” k�kR1 on Hn (M, R) by taking the infimumover all representatives. That is, we define

k�kR1 := inf

�kakR

1��a 2 Cn (M, R) , @a = 0, [a] = � 2 Hn (M, R)

for all � 2 Hn (M, R) . In particular, we write

kMkZ := [M]Z

Z

1 , kMkC := [M]C

C

1 , kMkQ := [M]Q

Q

1 . �

21

1 Simplicial Volume

Of course, we could also define a notion of volume using other homol-ogy theories based on chain complexes which can be equipped with some“norm” and whose top homology group (of oriented closed connected man-ifolds) contains a distinguished class.

The simplicial volume with integral coefficients obviously never vanishes.So it cannot be the same as the usual simplicial volume. But the integralsimplicial volume has an interesting connection with the Betti numbers ofthe corresponding manifold. However, the integral version is not very com-fortable to work with.

Lemma (1.46). Suppose M is an oriented closed connected n-dimensional manifoldand P t

j=1 aj ��j 2 Cn (M,Z) represents [M]Z . Then

Xk2N

bk(M) � t� 2n+1,

where bk(M) = rk Hk (M) is the k-th Betti number of M (which is the same asdimR Hk (M,R) and dimQ Hk (M,Q) ). In particular,

Xk2N

bk(M) � 2n+1 �kMkZ .

The proof is based on Poincare duality [Luck2, Example 14.28]. The follow-ing result is in the same direction, but is much harder to prove [Gromov2,page 220].

Theorem (1.47). For all k 2 R�1 and all n 2 N, there is a constant C(n, k)with the following property: If M is an n-dimensional oriented closed connectedRiemannian manifold with �k1 � sec(M) � k2 for some k1 � k2 > 0, then theBetti numbers of M can be estimated via

Xk2N

bk(M) � C(n, k1/k2)�kMk .

It is not hard to see that the real, complex, and rational simplicial volume allcoincide:

Lemma (1.48). Let M an oriented closed connected manifold of dimension n. Then

kMkC = kMk = kMkQ .

Proof. The inclusions C�(M,Q) � C�(M,R) � C�(M,C) yield

kMkC � kMk � kMkQ .

22

1.5 Generalisations

The inequality kMkC � kMk can be shown by considering the real part ofa representing cycle. The inequality kMk � kMkQ can be obtained via anapproximation of boundaries with real coefficients by boundaries with ra-tional coefficients. In both cases one has to use the fact that the fundamentalclass with rational or complex coefficients comes from a class in homologywith integral coefficients.

1.5.2 � More spaces

It is also possible to define a simplicial volume in more general settings[Gromov2, page 216]. For example, if M is a closed connected manifoldwhich is not orientable, we can define

kMk :=12� M

,

where M is the orientation double covering of M.

If M is not connected we can just sum up the simplicial volumes of the con-nected components.

If M is an oriented connected manifold without boundary which is not com-pact, we can use the fundamental class of M in locally finite homology todefine a (possibly infinite) simplicial volume.

If M is a manifold with boundary, then we can define the simplicial volumeas the seminorm of the relative fundamental class [M, @M].

Since the definition of the simplicial volume only makes use of the homolog-ical properties of manifolds, it is easy to find a definition of simplicial vol-ume for homology manifolds (with respect to real coefficients) or Poincarecomplexes, which are some kind of “homotopy manifolds.”

All these extensions preserve basic properties, such as homotopy invariance,multiplicativity etc.

1.5.3 � Functorial seminorms

Instead of the seminorm in singular homology with real coefficients inducedby the `1-norm, one could also use other seminorms. To ensure that theresulting invariant has useful properties, Gromov introduced the followingnotion [Gromov4, 5.34]:

23

1 Simplicial Volume

Definition (1.49). Suppose for each topological space X a seminorm k � k isgiven on H�(X,R). Such a seminorm is called functorial if for each contin-uous map f : X �! Y and all � 2 H�(X,R)

kH�(f ,R) (�)k � k�k. �

By Lemma (1.13) the seminorm k�k1 on homology with real coefficients in-duced by the `1-norm on the singular chain complex is functorial. Otherexamples are given in Gromov’s book [Gromov4, pages 302�316].

The proof of Corollary (1.16) of course shows that each functorial seminormis submultiplicative with respect to the degree of maps.

Definition (1.50). If k � k0 is a functorial seminorm, one can define an ana-logue of the simplicial volume as the value of k � k0on the fundamental class.

�

Then the same argument as in Corollary (1.18) shows that kSnk0must be zerofor all n 2 N>0 .

Using bounded cohomology, we will see in Corollary (2.28) that the semi-norm induced by the `1-norm vanishes on simply connected spaces. Gro-mov conjectured that this phenomenon occurs for all functorial seminorms[Gromov4, Remark in 5.35]:

Conjecture. Any functorial seminorm on singular homology with real coefficientsvanishes for simply connected spaces.

Until today, neither a proof nor counterexamples to this amazing conjectureare known.

24

2 Bounded Cohomology

Bounded cohomology is the functional analytic twin of singular cohomology – itis constructed similarly to singular cohomology from the singular chain complex,using the topological dual space instead of the algebraic dual space [Gromov2]. Thecorresponding duality on the level of (co)homology is explored in Section 2.2. Usingthis duality, it is possible to calculate the simplicial volume via bounded cohomol-ogy.

It is astonishing that the rather small difference between the definitions of singularand bounded cohomology lead to completely different characters of both theories.

Apart from the above geometric definition, there is also a notion of bounded co-homology of groups due to Ivanov based on techniques from homological algebra[Ivanov1], which we will sketch in Section 2.3. This algebraic approach is the sourceof the strength of bounded cohomology.

In Section 2.4, a key feature of bounded cohomology is discussed: the boundedcohomology of a topological space coincides with the bounded cohomology of itsfundamental group. Unfortunately, unlike singular cohomology bounded coho-mology fails to satisfy the excision axiom. Nevertheless, in many cases boundedcohomology can be calculated directly from special resolutions. For example, it canbe shown that bounded cohomology ignores amenable groups, implying that thebounded cohomology of spaces with amenable fundamental group vanishes.

In the last section, we will explain a special resolution for calculating the boundedcohomology of a group, which plays a crucial role in Section 4.3.

25


2.1 � Bounded cohomology of spaces

Bounded cohomology is the functional analytic twin of singular cohomol-ogy. It is constructed via the topological dual of the singular chain complexinstead of the algebraic one. The corresponding norm for singular cochainsis therefore the supremum norm:

Definition (2.1). Let X be a topological space and k 2 N. For a singularcochain f 2 Ck (X,R) the (possibly infinite) supremum norm is defined via

kfk 1 := sup�2Sk(X)

jf(�)j.

This induces a seminorm on Hk (X,R) by

8'2Hk(X,R) k'k 1 := inf�kfk 1

��f 2 Ck (X,R), Æ(f) = 0, [f ] = '

.

We writebCk (X) :=

�f 2 Ck (X,R)

�� kfk 1 < 1

.

for the vector space of bounded k-cochains. �

It is easy to see that the coboundary operator on the singular cochain com-plex Ck (X,R) restricts to bounded cochains, i.e.,

Æ�bCk (X)

�� bCk+1 (X).

Thus bC�(X) is a cochain complex.

Definition (2.2). Let X be a topological space and k 2 N. The k-th boundedcohomology group of X is defined by

bHk (X) :=ker

�ÆjbCk(X) :

bCk (X) �! bCk+1 (X)�

im�ÆjbCk�1 (X)

: bCk�1 (X) �! bCk (X)� . �

As for the simplicial volume, it is not immediately clear from the definitionthat there are spaces whose bounded cohomology does not vanish. An ex-ample of such a space is S1 _ S1 (Example (2.29)).

26

2.1 Bounded cohomology of spaces

Definition (2.3). Let X be a topological space and k 2 N. The supremumnorm on bCk (X) induces a seminorm on bHk (X) by

8'2 bHk(X) k'k 1 := inf

�kfk 1

��f 2 bCk (X), Æ(f) = 0, [f ] = '

. �

Except for some special cases it is not known whether this seminorm is anorm [Ivanov2], [Soma].

2.1.1 � Elementary properties of bounded cohomology

As we will see later, bounded cohomology is not as tame as singular co-homology because bounded cohomology fails to satisfy the excision axiom(so the usual divide-and-conquer approach does not work). But at leastbounded cohomology is natural, homotopy invariant and fulfils the dimen-sion axiom.

Lemma (2.4) (Naturality). Let X, Y be topological spaces and f : X �! Y acontinuous map. For each k 2 N, the map f induces a homomorphism

bHk (f) : bHk (Y)�! bHk (X)

[g] 7�!�c 7! g(f Æ c)

�

of operator norm at most 1. Here composition of f with elements of Ck (X,R) isdefined by linear extension of composition with elements of Sk (X).

Proof. Let g 2 bCk (Y)with Æ(g) = 0. Since g is bounded on Sk (Y), the cochain

Ck (X,R) �! Rc 7�! g(f Æ c)

is bounded on Sk (X). According to the definition of the coboundary opera-tor, this cochain is a cocycle. As the definition of bHk (f) ([g]) does not dependon the chosen representatives g, the map bHk (f) is indeed well-defined. It isclear that bHk (f) is a homomorphism of real vector spaces and that

bHk (f)

�[g]� 1� kgk1 .

Taking the infimum over all representatives yields bHk (f)

�[g]� 1� k[g]k1 .

27


Lemma (2.5) (Homotopy Invariance). The functor bounded cohomology is ho-motopy invariant, i.e., if f , g : X �! Y are homotopic maps of topological spaces,then the induced maps are equal:

bH� (f) = bH� (g) : bH� (Y)�! bH� (X).

Proof. Since f and g are homotopic maps, there is a chain homotopy

k�: C�(X,R) �! C�+1(Y ,R)

between C�(f ,R) and C�(g,R) [Bredon, Corollary IV 16.5], i.e.,

@Æ k�+1+ k�Æ@ = C�+1(f ,R) �C�+1(g,R).

This chain homotopy induces the map

h�: C�(Y ,R)�! C��1 (X,R)c 7�! (�1)�� cÆ k��1.

Obviously, h�maps bounded cochains to bounded cochains, thus it restrictsto

bh� := h�jbC�(Y):bC�(Y)�! bC��1 (X).

Since�Æ Æ bh�+ bh�+1ÆÆ

�(c)= Æ

�(�1)�� cÆ k��1

�+ bh�+1�(�1)�+1� cÆ@

�

= cÆ k��1 Æ@+ cÆ@Æ k�

= cÆ�C�(f ,R) �C�(g,R)

�

=�bC�(f) � bC�(g)

�(c)

holds for all c 2 bC�(Y), it follows that bh� is a cochain homotopy betweenbC�(f) and bC�(g). Hence we get bH� (f) = bH� (g).

Lemma (2.6) (Dimension Axiom). Bounded cohomology satisfies the dimensionaxiom, i.e.,

bH0 (�)�= R, and

8k2N�1bHk (�)= 0.

Proof. This is the same calculation as in singular cohomology.

28

2.2 Duality

2.2 � Duality

In this section the duality between the seminorms k�k1 and k�k1 on singu-lar homology and bounded cohomology respectively is explored. Using theHahn-Banach theorem we will see that the duality on the level of (co)chaincomplexes carries over to (bounded co)homology. This duality was discov-ered and applied by Gromov [Gromov2].

The duality principle (Theorem (2.7)) shows an important aspect of boundedcohomology: bounded cohomology can be used to compute the simplicialvolume. Since bounded cohomology is much better understood (in view ofthe techniques presented in Section 2.3) than the seminorm on homology,duality leads to interesting applications. For example, bounded cohomol-ogy can be used to give estimates of the simplicial volume of products andconnected sums of manifolds.

Moreover, duality will play a central role in the proof that measure homol-ogy and singular homology are isometrically isomorphic (see Section 4.3).

Theorem (2.7) (Duality Principle). Let X be a topological space, k 2 N and� 2 Hk (X,R).

1. Then k�k1 = 0 if and only if

8'2 bHk(X) h', �i = 0.

2. If k�k1 > 0, then

1k�k1

= inf�k'k 1

�� ' 2 bHk (X), h', �i = 1

.

The theorem makes use of the Kronecker product on bounded cohomol-ogy. Therefore, we recall the definition of the Kronecker product on singular(co)homology:

Definition (2.8). Let X be a topological space, R a commutative ring withunit and k 2 N. Furthermore, let f 2 Ck (X, R) be a cocycle and a 2 Ck (X, R)a cycle. Then the Kronecker product of [f ] 2 Hk (X, R) and [a] 2 Hk (X, R)is defined by

[f ], [a]�:= f(a). �

29


In fact, the relations between the coboundary and the boundary operatorimply that the Kronecker product is well-defined.

Definition (2.9). Let X be a topological space. The homomorphsim

cX : bH� (X) �! H�(X,R)

induced by the inclusion bC�(X) � C�(X,R) is called comparison map. �

In general, the comparison map is neither injective nor surjective: for S1

it is not injective, for S1 _ S1 it cannot be surjective (Corollary (2.28) andExample (2.29)).

Definition (2.10). Let X be a topological space, k 2 N, ' 2 bHk (X), and� 2 Hk (X,R). Then the Kronecker product of ' and � is defined to be

h', �i :=cX('), �

�. �

Proof (of Theorem (2.7)). Let � be represented by the cycle

a = X�2Sk(X)

a�� 2 Ck (X,R).

Assume that there is a ' 2 bHk (X), represented by the bounded cochainf 2 bCk (X), with h', �i = 1. Then, by definition,

1 = jh', �i j =

��f

�

X�2Sk(X)

a��

� �� X

�2Sk(X)

ja�j � jf(�)j

� kfk 1 �kak1 .

If we take the infimum over all representatives of ' and �, it follows that

1 � k'k 1 �k�k1 .

In particular, if k�k1 = 0, there can be no such '.

Now, let k�k1 > 0, and let a 2 Ck (X,R) be a cycle representing �. We de-note the closure in Ck (X,R) with respect to the norm k � k1 by �1. Thus thecondition kak1 � k�k1 > 0 implies

a /2 im�@ : Ck+1 (X,R) ! Ck (X,R)

�1.

30

2.2 Duality

Hence the Hahn-Banach theorem yields the existence of a continuous func-tional f : Ck (X,R) �! R with

f(a) = 1,f j

im�

@ : Ck+1(X,R)!Ck(X,R)�1 = 0,

f 1

=1

kak1.

The second property implies that f :=2 Ck (X,R) is a cocycle. Moreover,kfk 1 < 1, which shows f 2 bCk(X). By construction, we have

[f ], �

�=[f ], [a]

�= f(a) = 1

and

k[f ]k 1 � kfk 1 =1

kak1�

1k�k1

.

In particular, there exists a ' 2 bHk (X) with h', �i = 1 and k'k 1 � 1/ k�k1.Hence, if k�k1 > 0,

1k�k1

= inf�k'k 1

�� ' 2 bHk (X), h', �i = 1

.

Corollary (2.11). Let f : X �! Y be a continuous map such that the induced mapbH� (f) : bH� (Y)�! bH� (X) is an isometric isomorphism. Then

k�k1 = H�(f ,R) (�)

1

for all � 2 H�(X,R).

Proof. Since bH� (f) is an isometric isomorphism, we obtain

�k'k 1

��' 2 bHn (Y),

', Hn (f ,R) (�)

�= 1

=� bHn (f) (')

1

�� ' 2 bHn (Y),

bHn (f) ('), �

�= 1

=�k k 1

�� 2 bHn (X), h , �i = 1

.

Therefore application of Theorem (2.7) proves the Corollary.

31


2.2.1 � Bounded cohomology and simplicial volume

Application of the duality principle to the fundamental class yields a de-scription of the simplicial volume in terms of bounded cohomology:

Corollary (2.12). Let M be an oriented closed connected n-dimensional manifold.

1. Then kMk = 0 if and only if the comparison homomorphism

cnM : bHn (M) �! Hn (M,R)

is trivial.

2. If kMk > 0, then1

kMk= [M]R

1

,

where [M]R 2 Hn (M,R) denotes the image of the cohomological fundamen-tal class of M under the change of rings homomorphism.

Proof. By definition,[M]R , [M]R

�= 1. Furthermore,

Hn (M,R) �= R� [M]R .

Hence im(cnM ) 6= 0 if and only if [M]R 2 im(cn

M ).

1. Therefore the first part of Theorem (2.7) yields that kMk = k[M]Rk1 = 0if and only if cn

M is trivial.

2. Suppose kMk > 0. By the first part, cnM : bHn (M) �! Hn (M,R) �= R is

not trivial and hence surjective. Moreover,

k'k 1 = inf�kfk 1

��f 2 bCn (X), Æ(f) = 0, [f ] = ' in bHn (M)

� inf�kfk 1

��f 2 Cn (X,R), Æ(f) = 0, [f ] = cn

M (') in Hn (M,R)

= kcnM (')k 1

holds for all ' 2 bHn (M) . Using the surjectivity of cnM , we obtain (by the

second part of Theorem (2.7))

1kMk

=1

[M]R

1

= inf�k'k 1

��' 2 bHn (M) , h', [M]Ri = 1

� inf�kcn

M (')k 1

��' 2 bHn (M) ,

cn

M ('), [M]R�= 1

� inf�k k 1

�� 2 Hn (M,R) �= [M]R�R, h , [M]Ri = 1

= [M]R

1

.

32

2.2 Duality

In particular, [M]R

1

is finite. On the other hand, for each " 2 R>0 there isa cocycle f 2 Cn (M,R) with [f ] = [M]R in Hn (M,R) and

kfk 1 � [M]R

1+ " < 1.

Thus f 2 bCn (M) . Let ' 2 bHn (M) be the corresponding bounded cohomol-ogy class. Then

k'k 1 � kfk 1 � [M]R

1+ "

and

h', [M]Ri = hcnM ('), [M]Ri = h[f ], [M]Ri =

[M]R , [M]R

�= 1,

which implies

1kMk

=1

[M]R

1� k'k 1 �

[M]R

1+ ".

In Subsection 2.4.3 we will combine this result with the mapping theorem(Theorem (2.31)) to compute the simplicial volume in some special cases.

2.2.2 � The first bounded cohomology group

Using the result of Theorem (1.11) it is not hard to see that the first boundedcohomology group always vanishes:

Corollary (2.13). For all topological spaces X the group bH1 (X) is zero.

Proof. By Theorem (1.11) the seminorm on H1 (X,R) is trivial if X is arcwiseconnected. Since the singular homology of a space is the direct sum of thesingular homology groups of its path components, this result carries over toall spaces X.

Now let ' 2 bH1 (X) and let f 2 bC1 (X) be a cocycle representing '. Since theseminorm on H1 (X,R) is trivial (Theorem (1.11)), we get from Theorem (2.7)that

8�2H1(X,R) 0 = h', �i .

By the universal coefficient theorem,

H1 (X,R) �! hom�H1 (X),R

�

7�!�� 7! h , �i

�

33


is an isomorphism. Hence the comparison map c1X : bH1 (X) �! H1 (X,R)

must be trivial. Thus there is a cochain u 2 C0 (X,R) with

Æ(u) = f .

However, in general u is not bounded. Altering u, we construct a boundedcochain bu 2 bC0 (X) satisfying Æ(bu) = f : Let (X)i2I be the path componentsof X. For each i 2 I we choose a point xi 2 Xi. If x 2 X, then there is exactlyone i(x) 2 I such that x 2 Xi(x) . We define bu 2 C0 (X,R) to be the linearextension of

8x2X bu(x) := u(x) � u(xi(x)).

Since x and xi(x) are in the same path component, there is a path �x in X

joining these points. So��bu(x)

��=

��u(�x(0))� u(�x(1))

��

=��u(@(�x))

��

=��Æ(u)(�x)

��

=��f(�x)

��

� kfk 1 ,

which implies bu 2 bC0 (X). Furthermore we have for all � 2 S1 (X)

Æ(bu)(�)= bu(�(1)��(0))= bu(�(1))� bu(�(0))= u(�(1)) � u(xi(�(1))) � u(�(0)) + u(xi(�(0))).

Here i(�(1)) = i(�(0)) because �(1) and �(0) lie in the same path compo-nent. Hence

Æ(bu)(�)= u(�(1)) � u(�(0))= u(�@(�))

= Æ(u)(�)

= f(�).

This proves ' = [f ] = 0 in bH� (X).

2.2.3 � Products of manifolds

Bounded cohomology makes it possible to prove that the simplicial volumeis compatible with taking products (up to a constant) [Gromov2, page 218].

34

2.3 Bounded cohomology of groups

Theorem (2.14). For each n 2 N>0 there is a constant c(n) 2 R>0 such that thefollowing holds: If M and N are two oriented closed connected manifolds, then

kNk � kMk � kN �Mk � c(dim N + dim M)�kNk � kMk .

The proof makes use of explicit descriptions of the cup-product and thecross-products in singular (co)homology. In order to see the first inequal-ity, duality is applied. A detailed proof is given in the book of Benedetti andPetronio [Benedetti, Petronio, Theorem F.2.5]. The constant c(n) is related tothe number of n-simplices needed to triangulate the products �p ��n�p forall p 2 f1, : : : , ng.

2.3 � Bounded cohomology of groups

In his paper “Foundations of the Theory of Bounded Cohomology,” Ivanovdeveloped a transparent derived functor type approach to bounded coho-mology [Ivanov1]. Just like Ivanov, we will consider bounded cohomologyof discrete groups, but there is also a theory available for topological groups[Monod].

2.3.1 � Relatively injective resolutions

Similar to ordinary group cohomology, the bounded cohomology of a groupis based upon special kinds of resolutions. The basic objects are relativelyinjective bounded G-modules.

Definition (2.15). Let G be a discrete group.

� A bounded G-module is a Banach space V with a (left) G-action suchthat kg� vk � kvk for all g 2 G and all v 2 V .

� A G-morphism of bounded G-modules is a bounded G-equivariantlinear operator.

� An injective G-morphism i : V �! W between bounded G-modulesis said to be strongly injective if there is a bounded linear operator

35


s : W �! V satisfying sÆ i = id and ksk � 1 (but s does not need to beG-equivariant).

� A bounded G-module U is relatively injective if for any strongly in-jective G-morphism i : V �! W and any G-morphism � : V �! U ofbounded G-modules there is a G-morphism � : W �! U of boundedG-modules such that �Æ i = � and k�k � k�k. �

Good candidates for relatively injective bounded G-modules are mappingspaces:

Definition (2.16). Let G be a discrete group and let V be a Banach space. Wewrite B(G, V ) for the real vector space of bounded functions G �! V . Thisvector space admits a left G-action via

8f2B(G,V ) 8g2G g� f :=�h 7! f(hg)

�. �

Example (2.17). Let V be a Banach space and let G be a discrete group. Thesupremum norm turns B(G, V ) into a bounded G-module. Moreover, thisbounded G-module is relatively injective [Ivanov1, Lemma (3.2.2)]. �

Definition (2.18). Let G be a discrete group. A G-resolution of a boundedG-module V is a resolution of V via bounded G-modules and G-morphisms

0 V V0 V1 ��

Such a resolution is strong if there is a contracting homotopy, i.e., there is achain contraction (Kn)n2N consisting of (not necessarily equivariant) linearoperators with kKnk � 1 for all n 2 N. The above resolution is called rel-atively injective if the bounded G-modules (Vn)n2N are relatively injective.

�

Example (2.19). Let G be a discrete group. There always exists a strong rel-atively injective resolution of the trivial G-module R, the (inhomogeneous)standard resolution:

For n 2 N we define Vn := B(Gn,R). Equipped with the supremum normand the left G-action given by

g� f :=�(g1, : : : , gn) 7! f(g1, : : : , gn�1 , gn � g)

�

the module Vn becomes a relatively injective bounded G-module. The ho-momorphisms

d�1 : R �! V0

c 7�! (g 7! c)

36


and (for n 2 N)

dn : Vn�! Vn+1

f 7�!�(g0, : : : , gn+1) 7! (�1)n+1 � f(g1, : : : , gn+1)

+n

Xj=1

(�1)n�j � f(g0, : : : , gjgj+1, : : : , gn+1)�

turn (Vn)n2N into a strong relatively injective resolution [Ivanov1, (3.4)].Analogously to the classical case, one can also define the corresponding ho-mogeneous resolution. �

There is also a version of the fundamental theorem of homological alge-bra in the setting of bounded G-modules [Ivanov1, Lemma (3.3.2)], [Monod,Lemma 7.2.4 and 7.2.6]:

Theorem (2.20). Let G be a discrete group and let U and V be bounded G-modules.Suppose

0 V"V

V0v0

V1v1

��

is a strong resolution of V and

0 U"U

U0u0

U1u1

��

is a complex of relatively injective bounded G-modules. Then any G-morphismw : V �! U can be extended to a G-morphism (wn)n2N of complexes:

0 V"V

w

V0v0

w0

V1v1

w1

��

0 U "UU0

u0U1

u1��.

Any two such extensions are chain homotopic via a G-equivariant homotopy.

2.3.2 � Definition of bounded cohomology of groups

In view of Theorem (2.20), bounded cohomology of groups can be definedin the same way as usual group cohomology:

Definition (2.21). If G is a discrete group and V is a bounded G-module, wewrite V G for the set of G-fixed points. �

37


Definition (2.22). Let G be a discrete group and let

R V0v0

V1v1

��

be a strong relatively injective resolution of the trivial bounded G-moduleR.Then the bounded cohomology of G (with coefficients in R), bH� (G), is thecohomology of the induced complex

0 V G0

v0jV G

1v1j

��. �

Of course, one could also consider bounded cohomology of groups withmore general coefficients [Monod]. But in the following we will need onlybounded cohomology with real coefficients.

Corollary (2.23). Let G be a discrete group. Then any G-morphism between twostrong relatively injective resolutions of the trivial bounded G-moduleR by boundedG-modules extending the identity R �! R induces the same isomorphism on thecohomology of the G-fixed point sets.

In particular, the bounded cohomology bH� (G) is well-defined (up to canonical iso-morphisms).

Proof. This follows directly from Theorem (2.20). A detailed proof is givenby Ivanov [Ivanov1, Section (3.7)].

Each strong relatively injective resolution of the trivial G-module R givesrise to a seminorm on bH� (G). The topology on bH� (G) is independent of thechosen resolution, but the seminorm does depend on the resolution. How-ever, it is known that the standard resolution (compare Example (2.19)) in-duces the minimum of all these seminorms [Ivanov1, Theorem (3.6)], theso-called canonical seminorm.

As in the case of ordinary group cohomology, each group homomorphismf : G �! H induces a homomorphism bH� (f) : bH� (H) �! bH� (G) (for arigorous definition see [Ivanov1, Section (3.7)]). This turns bounded coho-mology of groups into a functor.

As one might suspect, bH� (G) �= bH� (BG) if BG is a model of the classifyingspace of G. In fact, much more holds, as is explained in Section 2.4.

38


2.3.3 � Amenability

An interesting facet of bounded cohomology is that it ignores a large classof groups – the amenable groups. This gives rise to interesting applicationsto the simplicial volume (see Subsection 2.4.3).

Definition (2.24). A group G is called amenable if the there is a G-invariantfunction m : B(G,R) �! R such that for all f 2 B(G,R)

infg2G

f(g) � m(f) � supg2G

f(g). �

Actually, this might seem a far-fetched definition. But amenable groups turnout to be quite interesting – for example they are connected with the Banach-Tarski paradox [Runde, Chapter 0].

In the above definition, m should be interpreted as a kind of special meanon G. Obviously, all finite groups are amenable – in this case m can be takenas integration over G with respect to the counting measure and dividing bythe number of elements in G. Furthermore, all Abelian groups are amenable[Runde, Example 1.1.5].

Ivanov shows that bounded cohomology cannot detect amenable groups[Ivanov1, Section 3.8]:

Theorem (2.25). Let A be a normal amenable subgroup of the discrete group G.Then the map bH� (G/A) �! bH� (G) induced by the canonical homomorphismG �! G/A is an isometric isomorphism.

Using the right invariant “mean” on A Ivanov proves that the standard reso-lution of the quotient group G/A also is strong relatively injective resolutionofR by bounded G-modules. Moreover, the induced canonical isomorphismturns out to be isometric.

The theorem shows that the free group Z � Z cannot be amenable: It can beshown that bH2 (Z� Z) is not even finitely generated. The “proof” of Brooks[Brooks] gives good intuition about this fact, but is not entirely correct – acorrect version can be found in the arcticles of Grigorchuk and Mitsumatsu[Grigorchuk], [Mitsumatsu].

39


2.4 � The mapping theorem

Perhaps surprisingly, the rather small differences between the definitionsof singular and bounded cohomology lead to significant differences in thebehaviour of both theories. For instance, Gromov and Ivanov showed thatthe bounded cohomology of spaces only depends on the fundamental group(Theorem (2.27)). More precisely, if f : X �! Y is a continuous map, there isa commutative diagram

bH� (Y)bH�(f)

�=

bH� (X)

�=

bH� (�1(Y))bH�(�1(f))

bH� (�1(X))

whose vertical arrows are canonical isometric isomorphisms. This diagramallows us to understand the bounded cohomology of spaces via Ivanov’stransparent axiomatic approach of Section 2.3.

2.4.1 � Bounded cohomology and the fundamental group

One proceeds in two steps to obtain the fact that bounded cohomology of aspace only depends on its fundamental group:

Theorem (2.26). Let X be a simply connected CW-complex with countably manycells. Then the bounded cohomology of X is canonically isometrically isomorphic tothe bounded cohomology of the one point space.

Theorem (2.27). Let X be a connected countable CW-complex. Then bH� (X) iscanonically isometrically isomorphic to bH�

��1(X)

�.

Actually, these theorems remain true if we drop the condition that the spacesmust be countable cellular. This situation is treated in Gromov’s originalexposition [Gromov2, Section 3.1].

40

2.4 The mapping theorem

Quite understandable proofs of the above theorems can be found in the pa-per of Ivanov [Ivanov1]. We will sketch these proofs: in the following, weview bCk (X) as the set of bounded functions Sk (X) �! R.

To prove the first theorem a contracting chain homotopy

R bC0 (X)K0 bC1 (X)

K1 ��K2

with kKnk � 1 for all n 2 N is constructed explicitly using “towers of con-nective covers.” Here it is needed that X is a countable CW-complex to en-sure that its homotopy groups are all countable [Lundell, Weingram, Theo-rem IV 6.1]. This implies that there is a model of the Eilenberg-Mac Lanespace K

��n(X), n

�which is a topological Abelian group Gn. Then there is

such a tower��

p3X3

p2X2

p1X1 = X

consisting of principal Gn-bundles pn : Xn+1 �! Xn. Ivanov shows that inthis situation for each n 2 N there is a chain map An : bC

�(Xn+1) �! bC�(Xn)

satisfyingAn Æ bH� (pn) = id and kAnk = 1.

Using these (An)n2N , the contracting homotopy K� can inductively be con-structed.

In order to prove the second theorem, the first theorem is applied to showthat (where p : eX �! X is the universal covering of X)

0 R bC0( eX) bC1( eX) ��

is a strong relatively injective resolution of the trivial bounded �1(X)-mod-uleR. Here the left �1(X)-action on bC�( eX) is given by the right �1(X)-actionon the universal covering space eX: for all f 2 bCk (X) and all g 2 �1(X)

g� f :=�� 7�! f(x 7! �(x)� g)

�.

Thus the bounded cohomology of the discrete group �1(X) can be computedas the cohomology of the complex

0 bC0( eX)�1(X) bC1( eX)�1(X) ��

The map bC�(X) �! bC��eX�

induced by p establishes an isomorphism be-tween bC�(X) and bC�( eX)�1(X) commuting with the differentials. Hence the

41


bounded cohomology of �1(X) coincides with the cohomology of the com-plex

0 bC0 (X)@ bC1 (X)

@ ��

which is the bounded cohomology of the space X. A more detailed calcula-tion reveals that the induced isomorphism is actually isometric.

Corollary (2.28). If X is a connected countable CW-complex with amenable funda-mental group, then bH� (X) �= bH� (�).

Example (2.29). The above relation between bounded cohomology and fun-damental groups also yields that bounded cohomology cannot satisfy ex-cision and that we cannot expect to find a cellular version of bounded co-homology: We have �1(S

1 _ S1) = Z � Z, and bH2 (Z� Z) is non-trivial (seeSubsection 2.3.3), but bH�

�S1��= bH� (�). �

Example (2.30). There is no interesting theory of characteristic classes forreal vector bundles in bounded cohomology: For each n 2 N, the fundamen-tal group of the classifying space BO(n) is amenable. Therefore the boundedcohomology bH� (BO(n)) coincides with the bounded cohomology of the onepoint space.

However, there are interesting results about the supremum norm of the ordi-nary Euler class of orientable flat vector bundles [Gromov2, page 231]. �

2.4.2 � The mapping theorem

If f : X �! Y is a continuous map, chasing through all the definitions andisomorphisms shows that bH� (f) = bH� (�1(f)) holds with respect to theisometric isomorphisms bH� (X) �= bH� (�1(X)) and bH� (Y) �= bH� (�1(Y)) ofTheorem (2.27). Thus combining Theorem (2.27) and Theorem (2.25) yields:

Theorem (2.31) (Mapping Theorem for Bounded Cohomology). Let X andY be countable CW-complexes and let f : X �! Y be a continuous map. If thehomomorphism �1(f) : �1(X) �! �1(Y) is surjective and its kernel is amenable,then the induced homomorphism

bH� (f) = bH� (�1(f)) : bH� (Y)�! bH� (X)

is an isometric isomorphism.

42

2.4 The mapping theorem

By Corollary (2.11), in the situation of Theorem (2.31), the homomorphismH�(f ,R) then also preserves the seminorm in homology.

Again, the mapping theorem remains true if we drop the condition for thespaces to be countable cellular [Gromov2, Section 3.1].

2.4.3 � Applications to simplicial volume

In particular, we can apply the mapping theorem to obtain the followingresults about the simplicial volume:

Corollary (2.32). Let M be an oriented closed connected manifold of positive di-mension n, and let G be its fundamental group.

1. If f : M �! BG denotes the classifying map, then

kMk = kHn (f ,R) ([M]R)k1 .

2. Moreover, if G is amenable, then kMk = 0.

Proof. 1. By construction, �1(f) is an isomorphism. Hence the mapping the-orem yields that bHn (f) is an isometric isomorphism. Now Corollary (2.11)shows kMk = kHn (f ,R) ([M]R)k1.

2. Corollary (2.28) implies that bHn (M) vanishes. Hence

kMk = [M]R

1 = 0,

by duality (the first part of Theorem (2.7)).

A nice argument from covering theory yields the following result [Luck2,Exercise 14.11]:

Corollary (2.33). Let M and N be oriented closed connected manifolds of the samedimension and let f : M �! N be a continuous map of degree 1. Then �1(f) issurjective. If the kernel of �1(f) is amenable, then

kMk = kNk .

Example (2.34). However, the simplicial volume does not only depend onthe fundamental group of the manifold: Let F2 be the oriented closed con-nected surface of genus 2. Then F2 and F2 � S2 have the same fundamentalgroup, but

kF2k= 4 and kF2 � S2k = 0

by Example (1.24) and Theorem (2.14) respectively. �

43


Furthermore, one can derive from the mapping theorem that the simplicialvolume is additive with respect to connected sums.

Theorem (2.35). Let M and N be two oriented closed connected manifolds of thesame dimension n � 3. Then

kM # Nk = kMk + kNk .

Gromov’s proof of this theorem combines beautiful geometric ideas of sim-plicial “tree-like” complexes with the mapping theorem (applied to the ev-ident map M # N �! M _ N). However, his exposition in “Volume andBounded Cohomology” [Gromov2, Section 3.5] is rather short.

2.5 � A special resolution

We close this chapter with the discussion of a special resolution which com-putes bounded cohomology. This resolution will be one of the ingredientsneeded to show that singular homology and measure homology are isomet-rically isomorphic (see Section 4.3).

Definition (2.36). Let X be an arcwise connected space with universal cov-ering eX, and let k 2 N.

� Then �1(X) acts from the left on the vector space map(eXk+1,R) of con-tinuous functions eXk+1 �! R via

g� f :=�(x0, : : : , xk) 7! f(x0 � g, : : : , xk � g)

�

for all f 2 map( eXk+1,R) and all g 2 �1(X).

� The subset of bounded functions in map(eXk+1,R) is denoted by F�(X)

and we use the abbreviation Ik (X) for the functions in Fk (X) which areinvariant under the above �1(X)-action. �

How can we turn F�(X) into a cochain complex? The vector space bCk( eX)

can be identified with the space of bounded functions Sk( eX) �! R (underthis identification, the norm k�k1 just becomes the supremum norm).

44

2.5 A special resolution

Now Fk (X) can be viewed as a subspace of bCk( eX), namely as the space ofthose bounded functions Sk( eX) �! R which only depend on the vertices ofthe simplices (and are continuous in the vertices). In Gromov’s terminologythose functions would be called “straight bounded continuous cochains”[Gromov2, Section 2.3].

It is clear that the coboundary operator on bC�( eX) restricts to F�(X) and thatthe operations of �1(X) on F�(X) and bC�( eX) are compatible with the aboveinclusion map. This makes F�(X) a subcomplex of bC�( eX). In other words,the homomorphism

uk : Fk (X) �! bCk( eX)

f 7�!�� 7! f(�(e0), : : : , �(ek))

�

is a �1(X)-equivariant cochain map.

Theorem (2.37). Let X be a connected countable CW-complex and let eX be itsuniversal covering space.

1. Then the homomorphism

": R �! F0 (X) = map( eX,R)c 7�! (x 7! c)

turns the cochain complex F�(X) into a strong relatively injective resolutionby bounded �1(X)-modules of the trivial �1(X)-module R.

2. Hence the cohomology of I�(X) is canonically isomorphic to bH� (�1(X)) .Moreover, under this isomorphism the supremum norm on F�(X) inducesthe canonical seminorm on bH� (�1(X)) .

Corollary (2.38). Let X be a connected countable CW-complex. Then H��I�(X)

�

and bH� (X) are isometrically isomorphic (by Theorem (2.27)).

A proof of the theorem is given in Monod’s book [Monod, Theorem 7.4.5].However, Monod does not explicitly write down a cochain map

I�(X) �! bC�(X)

inducing this isometric isomorphism. Since we need a concrete descriptionof this map in Section 4.3, we will now construct such a cochain map withthe help of the uniqueness part of Theorem (2.20).

45


As mentioned in Section 2.4, the bounded cohomology groups bH� (�1(X))

can be computed via the strong relatively injective resolution bC�( eX). There-fore we will compare the two complexes I�(X) and bC�( eX): The “inclusion”map u� from above fits into the ladder

0 R"

id

F0 (X)

u0

F1 (X)

u1

��

0 R "bC0( eX) bC1( eX) ��.

Now Corollary (2.23) implies that (after taking the �1(X)-fixed points) u�

induces the canonical isomorphism

H��I�(X)

��! H�

�bC�( eX)�1(X)

�

(which is isometric by Theorem (2.37)). As mentioned in Section 2.4, theisometric isomorphism

bp�: bH� (X) �! H��bC�( eX)�1(X)

�

is induced by the universal covering map p : eX �! X. Thus the desired iso-metric isomorphism H�

�I�(X)

��! bH� (X) is the homomorphism H�(w�)

making the triangle

H��I�(X)

� H�(w�)

H�(uj�)

bH� (X)

bp�

H��bC�( eX)�1(X)

�

commutative (where uj� denotes the restriction of u� to the �1(X)-fixedpoints).

The inverse of bp� on the level of cochains can be described as follows: for allf 2 bCk( eX)�1(X) and � 2 Sk (X) we define

�vk(f)

�(�) := f(e�),

where e� is some lift of � with respect to p. This definition does not dependon the chosen lift since f is �1(X)-invariant and any two lifts of � only differby multiplication with an element in �1(X) [Massey, Lemma V 8.1]. Sincef is bounded, vk(f) is bounded on Sk (X). Therefore linear extension yieldsvk(f) 2 bCk (X).

46

2.5 A special resolution

Covering theory implies that for all � 2 Sk (X) and all j 2 f0, : : : , kg each liftof the j-th face of � is also the j-th face of some lift of � . Hence

v�: bC�( eX)�1(X) �! bC�(X)

is a cochain map. By construction, it is the inverse of the cochain mapbC�(X) �! bC�( eX)�1(X) induced by the universal covering map p.

Therefore, the composition

w� := v�Æ uj�: I�(X) �! bC�(X)

induces the canonical isometric isomorphism H�(I�(X)) �! bH� (X). Bydefinition, for all f 2 Ik (X) and all � 2 Sk (X) we obtain

�wk(f)

�(�) = f

�e�(e0), : : : , e�(ek)

�,

where e�is any lift of � . This is the description we will use in Section 4.3.

47


48

3 Measure Homology

Measure homology is a generalisation of singular homology in the following way:singular chains with real coefficients are viewed as signed measures on the spaceof singular simplices whose mass is concentrated in finitely many points. In themeasure homology chain complex also more complicated measures are allowed.

Measure homology was introduced in Thurston’s lecture notes in 1979 [Thurston,page 6.6]. He already claimed that measure homology and singular homologyshould coincide. More extensive accounts are the papers of Zastrow [Zastrow] andHansen [Hansen] and the book of Ratcliffe [Ratcliffe].

The motivation for the introduction of measure homology originates from the factthat measure homology can be used to calculate the simplicial volume, hence giv-ing more room for constructions such as smearing (cf. Chapter 5). This smearingconstruction will be the key to the proof of the proportionality principle.

In Section 3.1, the technical background needed for the definition of measure ho-mology is presented – in particular, a feasible topology on the set of smooth sin-gular simplices will be introduced. In Section 3.2 measure homology is defined.The basic properties of measure homology are listed in Section 3.3, and there is alsogiven a proof for the compatibility of measure homology with colimits. The nextchapter is devoted to the proof that measure homology and singular homology areisometrically isomorphic.

49

3 Measure Homology

3.1 � Prelude

In this section the basic objects involved in the definition of measure ho-mology are examined. The first subsection introduces signed measures. Inthe later subsections, topologies for mapping spaces are provided and sometechnicalities concerning smooth maps on standard simplices are discussed.

3.1.1 � Signed measures

Signed measures are a slight generalisation of positive measures. As thename suggests, signed measures can also take negative values. The gener-alisation is quite straightforward; however, some additional care has to betaken when defining null sets or integrability with respect to a signed mea-sure. The main advantage of signed measures is that the set of all finitesigned measures on a given measurable space is a vector space over the realnumbers.

Definition (3.1). Let (X,A) be a measurable space.

� A map � : A �! R [ f1,�1g is called a signed measure if �(;) = 0,not both 1 and �1 are contained in the image of �, and � is �-additive.

� A null set of a signed measure � is a measurable set A with �(B) = 0for all B 2 A with B � A.

� A set A 2 A is called �-positive if �(B) � 0 for all B 2 A with B � A.Analogously, �-negative sets are defined.

� A determination set of � is a subset D of X such that each measurableset contained in the complement of D is a �-null set. �

Of course, each ordinary positive measure is also a signed measure. If �

and � are two positive measures, at least one of them being finite, then ��

and � � � are two signed measures. In fact, these are the only examples sinceeach signed measure can be written as such a difference.

50

3.1 Prelude

Theorem (3.2) (Hahn Decomposition). Let � be a signed measure on the mea-surable space (X,A). Then there is a decomposition X = P [ N where P is a�-positive set and N is a �-negative set.

This decomposition is essentially unique: if X = P 0 [N0 is another decompositionin a �-positive and a �-negative set, then the symmetric difference of P and P 0

(which equals the symmetric difference of N and N 0) is a �-null set.

A proof is, for example, given in Elstrodt’s textbook on measure theory andintegration [Elstrodt, Satz 1.8 in Kapitel VII].

Definition (3.3). Let � be a signed measure on (X,A) and X = P [ N itsHahn decomposition.

� The (positive) measure

�+ : A �! R�0 [ f1g

A 7�! �(A \P)

is called positive variation of �. Analogously, the restriction �� of�� to the set N is called negative variation of �. The decomposition� = �+ � �� is the Jordan decomposition of �.

� The variation of � is defined as the sum

j�j := �+ + ��

and the total variation of � is given by

k�k := j�j(X) 2 R�0 [ f1g. �

Since the Hahn decomposition is unique up to null sets, the positive and thenegative variation of a signed measure are indeed well-defined. Moreover,the sum �+ + �� makes sense because at least one of these two measuresmust be finite.

Sometimes it is useful to have another description of the total variation:

Lemma (3.4). Let (X,A) be a measurable space and � a signed measure on (X,A).Then

k�k = supA2A

�(A) � infB2A

�(B).

51

3 Measure Homology

Proof. Let X = P [N be the Hahn decomposition of �. By definition,

k�k = j�j(X) = �(P) � �(N).

This clearly shows “�”. On the other hand, for all A, B 2 A

�(A)� �(B) = �+(A) � �� (A)� �+(B) + �� (B)

� �+(A) + �� (B)

� �+(X) + �� (X)

= j�j(X)

= k�k

since �+ and �� are positive measures.

Definition (3.5). Let X be a topological space. The Borel �-algebra is the�-algebra on X generated by all open sets of X. �

Definition (3.6). Let (X,A) be a measurable space and f : (X,A) �! R ameasurable function (with respect to the Borel �-algebra on R). If � is asigned measure on (X,A), we call f integrable with respect to � if f is in-tegrable with respect to the positive measures �+ and �� in the usual sense.The integral of f with respect to � is then defined as

Z

f d� :=

Z

f d�+ �

Z

f d�� . �

Lemma (3.7). Let X be a topological space. The set of all signed measures on theBorel �-algebra of X possessing a compact determination set and finite total varia-tion is a real vector space. The total variation turns this vector space into a normedvector space.

Proof. Let � and � be two such measures and let � 2 R. Obviously, � � �again is a signed measure. Each determination set of � is a determinationset of �� and

k�� k= j�j � k�k .

Since both � and � have finite total variation, their value on each Borel setis finite. Hence the sum � + � is well-defined and again a signed measure.Furthermore, if D� and D� are (compact) determination sets of � and � re-spectively then D� [D� is a (compact) determination set of �+ �.

52

3.1 Prelude

Using Lemma (3.4) we get (where A is the Borel �-algebra on X)

k�+ �k= supA2A

��(A) + �(A)

�� inf

B2A

��(B) + �(B)

�

� supA2A

�(A) + supA2A

�(A) � infB2A

�(B) � infB2A

�(B)

= k�k + k�k.

In particular, k�+ �k is finite.

Definition (3.8). Let f : X �! Y be a Borel function, i.e., measurable withrespect to the Borel �-algebra. Moreover, let � be a signed measure on theBorel �-algebra of X. Then � induces a signed measure �f on Y given by

�f (A) := ��f�1 (A)

�

for all Borel subsets A � Y . �

Lemma (3.9). Let f : X �! Y be a continuous function and let � be a signed mea-sure on the Borel �-algebra of X possessing a compact determination set and finitetotal variation. Then the induced measure �f also has a compact determination setand finite total variation.

Proof. It is easy to see that continuous functions are Borel functions since theBorel �-algebra is generated by the open sets. Hence �f is well-defined.

If D is a compact determination set of �, then the image f(D) of course is adetermination set of �f . Since f is continuous, f(D) is compact.

Writing AX and AY for the Borel �-algebras on X and Y respectively wecompute the total variation of �f via Lemma (3.4) as follows

�f

= sup

A2AY

�f (A)� infB2AY

�f (B)

= supA2AY

��f�1 (A)

�� inf

B2AY��f�1 (B)

�

� supA2AX

�(A)� infB2AX

�(B)

= k�k .

Hence �f also has finite total variation.

53

3 Measure Homology

3.1.2 � Mapping spaces

Since we want to define measure chains as signed measures on some setof singular simplices, we have to introduce an appropriate topology on thecorresponding mapping space. Unfortunately, the compact open topologyis too coarse for our purposes. So we need another topology – the so-calledC1-topology defined on the space of smooth maps of smooth manifolds –,which takes into account the smooth structure.

Definition (3.10). Let X and Y be two topological spaces and map (X, Y)theset of continuous maps X �! Y .

� If K � X is compact and U � Y is open, we write

UK := ff 2 map (X, Y)j f(K) � Ug.

� The compact open topology on the mapping space map (X, Y) is thetopology with the subbase

fUK jK � X compact, U � Y openg. �

Remark (3.11). In particular, for all x 2 X the evaluation map

map (X, Y)�! Y

f 7�! f(x)

is continuous.

Definition (3.12). Let M and N be two smooth manifolds without bound-ary. As sets map1 (M, N) and mapco

1 (M, N) are just the sets of smoothmaps M �! N .

� The topology on mapco1 (M, N) � map (M, N) is the subspace topology

given by the compact open topology on map (M, N) .

� The topology on map1 (M, N) , called C1-topology, is the topologyconsisting of all inverse images of the form T�1 (U) where U is an opensubset of map (TM, TN)and T is the differential:

T : map1 (M, N) �! map (TM, TN)

f 7�! Tf .

I.e., the injective map T : map1 (M, N) �! map (TM, TN) is turnedinto a homeomorphism onto its image T

�map1 (M, N)

�. �

54

3.1 Prelude

More information about the C1-topology and similar topologies can be foundin Hirsch’s book [Hirsch1, Chapter 2]. The above description is taken from apaper of Hirsch [Hirsch2, page 244].

Remark (3.13). The compact open topology is the same as the topology ofuniform convergence on compact subsets [Dugundji, Theorem XII 7.2]. Sim-ilarly, the C1-topology is the topology of uniform convergence on compactsubsets not only of the function itself but also of its differential. �

Lemma (3.14). Let M and N be two smooth manifolds without boundary. Thenthe C1-topology on map1 (M, N) is finer than the compact open topology.

Proof. Denoting the zero section M ,! TM by i and the bundle projec-tion TN �! N by �, we get for all compact sets K � M and all open setsU � N that

T�UK \map1 (M, N)

�

=�

Tf : TM �! TN�� f 2 UK \map1 (M, N)

=�

Tf : TM �! TN�� f 2 map1 (M, N) , Tf(i(K)) � ��1 (U)

=�

g 2 map (TM, TN)�� g(i(K)) � ��1 (U)

\T

�map1 (M, N)

�.

Since i and � are continuous, i(K) � TM is compact and ��1 (U) � TN isopen. Thus the set UK \map1 (M, N) is open in map1 (M, N) .

Lemma (3.15). 1. Let X, Y , and Z be topological spaces and f 2 map (X, Y),g 2 map (Y , Z). Then the maps

map (Y , Z) �! map (X, Z)

h 7�! hÆ f

and

map (X, Y)�! map (X, Z)

h 7�! gÆ h

are continuous.

2. Let L, M, N be smooth manifolds without boundary and f 2 map1 (L, M) ,g 2 map1 (M, N) . Then the maps

map1 (M, N) �! map1 (L, N)

h 7�! hÆ f

55

3 Measure Homology

and

map1 (L, M) �! map1 (L, N)

h 7�! g Æ h

are continuous.

Proof. 1. This can easily be seen using the definition of the compact opentopology.

2. Applying the first part to Tf 2 map (TL, TM)and Tg 2 map (TM, TN)proves the second part.

3.1.3 � “Smooth” maps on standard simplices

Since the standard simplex �k is quite angular, some care has to be takenwhen defining the set of all smooth maps �k �! M, where M is somesmooth manifold without boundary. The elements of map1 (�

k, M) shouldbe thought of as being smooth maps on some slightly larger set without“corners.” More precisely:

Definition (3.16). Let M be a smooth manifold without boundary and k 2 N.

� The k-dimensional hyperplane in Rk+1 spanned by the vertices of thestandard k-simplex �k will be denoted by�k. We then write �k for theset of all open sets V � �k (in the subspace topology) containing �k.

� A map f : �k �! M is smooth if there is a V 2 �k and a smooth mapF 2 map1 (V , M) such that

F j�k = f .

The pair (F , V ) is called an extension of f .

� A map f : M �! �k is called smooth if the composition iÆ f is smooth,where i is an affine isomorphism �

k �! Rk composed with the inclu-sion �k ,! �k. We also use this terminology if M happens to be astandard simplex and f is smooth in the above sense.

� We write map1 (�k, M) and map1 (M,�k) for the set of all smooth

maps �k �! M and the set of all smooth maps M �! �k, respec-tively. �

56

3.1 Prelude

Remark (3.17). If M is a smooth manifold without boundary and k 2 N thenwe obviously get an inclusion map1 (�

k, M) ,! Sk (M) . �

The mapping space map(�k, M) can be topologised as follows:

Definition (3.18). Let M be a smooth manifold without boundary or a stan-dard simplex, and let k 2 N. We use the notation mapco

1 (�k, M) for the setmap1 (�

k, M) � map(�k, M) endowed with the subspace topology. �

However, we will also need some kind of C1-topology on map1 (�k, M). So

we introduce the differential of a smooth map �k �! M:

Definition (3.19). For k 2 Nwe use the notation

T�k := �k �Rk.

Moreover, if V 2 �k, we will always presuppose the canonical identification

TV = V �Rk

of the tangent bundle of the smooth k-manifold V . �

Definition (3.20). Let M be a smooth manifold without boundary or a stan-dard simplex, and let k 2 N. For a smooth map f 2 map1 (�

k, M) we defineits differential by

Tf := TF jT�k: T�k �! TM

where (F , V ) is some extension of f . �

This differential is indeed well-defined:

Lemma (3.21). Let M be a smooth manifold, k 2 N, and f 2 map1 (�k, M).

Suppose (F1, V1) and (F2, V2) are two extensions of f . Then

TF1j��11 (�k) = TF2j��1

2 (�k)

where �1 : TV1 �! V1 and �2 : TV2 �! V2 are the bundle projections.

Proof. Since F1 and F2 are smooth, their differentials TF1 and TF2 are contin-uous. Furthermore,

TF1j��11 ((�k)Æ) = TF2j��1

2 ((�k)Æ)

57

3 Measure Homology

since F1 and F2 extend f . For each (x, v) 2 ��11 (�k) = ��1

2 (�k) there is asequence (xn)n2N � (�k)Æsuch that

x = limn!1

xn,

which implies limn!1 (xn, v) = (x, v) in ��11 (�k) = ��1

2 (�k). Hence weconclude

TF1(x, v)= TF1�

limn!1

(xn, v)�= lim

n!1TF1(xn, v)

= limn!1

TF2(xn, v) = TF2�

limn!1

(xn, v)�

= TF2(x, v).

Having defined the differential of a smooth map �k �! M, we are now ableto introduce the C1-topology on map1 (�

k, M):

Definition (3.22). Let M be a smooth manifold without boundary or a stan-dard simplex, and let k 2 N. The C1-topology on map1 (�

k, M) is the topol-ogy induced by the compact open topology on map(T�k, TM) pulled backby the differential (as defined in Definition (3.20))

T : map1��k, M

��! map

�T�k, TM

�

f 7�! Tf . �

Lemma (3.23). Let M be a smooth manifold without boundary or a standard sim-plex, and let k 2 N. The topology of map1 (�

k, M) is finer than the topologyof mapco

1 (�k, M).

Proof. We can use literally the same proof as in Lemma (3.14).

Remark (3.24). In the same way we can derive statements corresponding tothe second part of Lemma (3.15). �

The following two rather technical results are needed for the proof that mea-sure homology is compatible with colimits.

Lemma (3.25). Let M be a smooth manifold without boundary and N � M anopen subset. Let k 2 N.

1. Then the mapj : map1

��k, N

��! map1

��k, M

�,

induced by the inclusion N ,! M, is continuous, injective, and open.

58

3.1 Prelude

2. Let im j := j�map1 (�

k, N)�� map1 (�

k, M) (endowed with the subspacetopology). Then the inverse im j �! map1 (�

k, N) of j is continuous.

Proof. 1. Since the inclusion N ,! M is smooth, continuity of j follows fromRemark (3.24). The fact that j is injective is clear because N is Borel in M.

The map j is open: Namely, since j is injective, taking the image of j iscompatible with intersections. Hence it suffices to show that j maps setsof the form T�1 (UK) with open U � TN and compact K � T�k to open setsin map1 (�

k, M). By definition,

j�T�1 (UK)

�

= j��

f 2 map1 (�k, N)

��Tf(K) � U

�

=�

iÆ f��f 2 map1 (�

k, N), Tf(K) � U

=�

f 2 map1 (�k, M)

��Tf(K) � U

\�

f 2 map(�k, M)��f(�k) � N

where i : N ,! M is the inclusion. Since N is open in M, the set U must beopen in TM. Thus the first set is open in map1 (�

k, M). Lemma (3.23) showsthat the second set is also open in map1 (�

k, M). Thus j is open.

2. This is an immediate consequence of the first part.

Corollary (3.26). Let M be a smooth manifold without boundary and let (Uk)k2Nbe an ascending family of open sets covering M. Let j 2 N. For each k 2 Nthe map map1 (�

j, Uk) �! map1 (�j, M) induced by the inclusion Uk ,! M

is denoted by jk. Then�jk(map1 (�

j, Uk))�

k2Nis an ascending open covering

of map1 (�j, M).

Proof. It is clear that this family is ascending. By Lemma (3.25) all the im-ages jk

�map1 (�

j, Uk)�

are open in map1 (�j, M).

Moreover, map1 (�j, M) is covered by

�jk(map1 (�

j, Uk))�

k2N: namely, if

f 2 map1 (�j, M), then the image f(�j) � M is compact. Since (Uk)k2N is

an ascending open covering of M, there is some k 2 N such that f(�j) � Uk.If (F , V ) is an extension of f , then V 0 := F�1 (Uk) is an open subset of V con-taining �j, and the pair

�F jV 0, V 0

�shows that there is an f 2 map1 (�

j, Uk)satisfying

f = jk(f ).

59

3 Measure Homology

3.2 � Definition of measure homology

Measure homology is a generalisation of singular homology in the follow-ing sense: Let X be a topological space, k 2 N, and let S � map

��k, X

�be

some set of simplices. The idea of measure homology is to think of a singu-lar chain P�2S a�� with real coefficients as a signed measure on S havingthe mass a� on the set f�g. The measure chain complex will consist of allsigned measures on S (modulo some finiteness condition). In particular, cer-tain “infinite” singular chains are allowed. Thus the measure chain complexis larger than the singular chain complex and hence gives more room for cer-tain constructions such as smearing (see Section 5.4). The other side of thecoin is that it is quite hard to get a geometric intuition of more complicatedmeasure chains.

Depending on the choice of the mapping space S (and its topology) there aretwo main flavours of measure homology:

� One for general (pairs of) spaces using the compact open topology onthe set of all singular simplices, and

� one for (pairs of) smooth manifolds using the C1-topology on the set ofsmooth singular simplices.

However, our application of measure homology in Chapter 5 compels us touse the smooth version. So we will only introduce this version in detail.

Measure homology was introduced by Thurston [Thurston, page 6.6]. Somebasic properties of smooth measure homology are provided in Ratcliffe’sbook [Ratcliffe, x11.5]. A thorough treatment of measure homology for gen-eral spaces is given by Zastrow and Hansen [Zastrow], [Hansen]. In The-orem 3.4 of Zastrow’s paper a way is described how to derive the corre-sponding results and proofs for the smooth theory. Just as Zastrow, we referto those adapted statements of his paper by numbers such as, e.g., 4.5diff in-stead of 4.5. Unfortunately, it is probably more transparent to provide a newproof for the compatibility with colimits instead of adapting Hansen’s result[Hansen, Proposition 5.1]. Therefore we give a full proof at the end of thischapter.

60

3.2 De�nition of measure homology

Definition (3.27). Let M be a smooth manifold without boundary and k 2 N.

� The k-th measure chain group, denoted by Ck (M) , is the R-vectorspace of all signed measures on the Borel �-algebra of the mappingspace map1 (�

k, M) possessing a compact determination set and finitetotal variation. Its elements are called measure k-chains.

� For each j 2 f0, : : : , k+ 1g the inclusion @j : �k �! �k+1 of the j-th face

induces by Remark (3.24) and Lemma (3.9) a homomorphism (whichwe will also denote by @j)

@j : Ck+1 (M)�! Ck (M)

� 7�! �(�7!�Æ@j)

We then define the boundary operator of measure chains by

@ : Ck+1 (M)�! Ck (M)

� 7�!k+1

Xj=0

(�1)j �@j(�). �

Without the condition on the variation, we would not get a vector space(since signed measures cannot be added in general). The compact deter-mination sets play in some way the role of the compactness of the standardsimplex (as can be seen in Lemma (3.45)) and the finiteness of singular chainsin singular homology.

Just as in singular homology theory, we suppress the dependency of @ on k

in the notation.

Lemma (3.28). Let M be a smooth manifold without boundary and let k 2 N. Then�C�(M) , @

�is a chain complex.

Proof. This is shown in [Zastrow, Corollary 2.9diff] and [Ratcliffe, Lemma 1in x11.5].

Statements like the previous lemma are proved by Zastrow, using a moregeneral framework which permits to translate some properties of singularhomology to the setting of measure homology [Zastrow, Theorem 2.1diff].

Due to the above lemma, we can naturally define a homology theory:

61

3 Measure Homology

Definition (3.29). Let M be a smooth manifold without boundary and k 2 N.The R-vector space

Hk (M) :=ker

�@ : Ck (M) �! Ck�1 (M)

�

im�@ : Ck+1 (M) �! Ck (M)

�

is called the k-th measure homology group of M. �

In view of the goal to compute the simplicial volume via measure homology,we introduce a norm on measure homology using the total variation:

Definition (3.30). Let M be a smooth manifold without boundary and letk 2 N. The total variation k�kinduces a seminorm on Hk (M) as follows:For all � 2 Ck (M) we define

k�kmh := inf�k�k

�� 2 Ck (M) , @(�) = 0, [�] = �

. �

Just like in singular homology, there is also a notion of relative homologygroups:

Definition (3.31). Let M be a smooth manifold without boundary and letN be a smooth submanifold without boundary. Due to Remark (3.24) andLemma (3.9), the inclusion N ,! M induces a homomorphism C�(N) �!C�(M) , which is injective since N is measurable. Hence for k 2 N we candefine the k-th relative measure chain group as the quotient vector space

Ck (M, N) :=Ck (M)

Ck (N). �

The boundary operator on the absolute measure chain groups clearly in-duces a boundary operator on the relative measure chain groups, turningthem into a chain complex.

Definition (3.32). Let M be a smooth manifold without boundary and letN � M be a smooth submanifold without boundary. For each k 2 N thek-th relative measure homology group is defined to be the quotient

Hk (M, N) :=ker

�@ : Ck (M, N) �! Ck�1 (M, N)

�

im�@ : Ck+1 (M, N) �! Ck (M, N)

�. �

Remark (3.33). Let M be a smooth manifold without boundary. Since thereis only one measure on the empty set (the zero-measure), we get C�(;) = 0.This yields canonical isomorphisms

C�(M) �= C�(M, ;) ,H�(M) �= H�(M, ;) . �

62

3.3 Basic properties of measure homology

3.3 � Basic properties of measure homology

In the category of smooth manifolds without boundary and smooth mapsmeasure homology behaves like singular homology with real coefficients;actually, most of the proofs are based on the same ideas as the correspondingproofs for singular homology theory.

Lemma (3.34) (Dimension Axiom). Measure homology satisfies the dimensionaxiom: For all n 2 N

Hn (�)�=

(R if n = 0,0 otherwise.

Proof. It is easy to see that the chain complexes�C�(�), @

�and

�C�(�,R), @

�

are the same because for each k 2 N there is just one map �k �! � (whichis smooth). Thus

H�(�)�= H�(�,R)

which proves the lemma.

Lemma (3.35) (Naturality). Each smooth map f : M �! N between smoothmanifolds without boundary induces a homomorphism

H�(f) : H�(M) �! H�(N)

in measure homology. This turns H� into a functor.

Proof. Since f is smooth, we get for each k 2 N an induced map

map1��k, M

��! map1

��k, N

�

�7�! f Æ� ,

which is continuous by Remark (3.24). Hence Lemma (3.9) shows that weobtain an induced map

Ck (M) �! Ck (N)

� 7�! �(�7!fÆ�)

which obviously is a homomorphism. This yields a chain map C�(M) �!C�(N) [Zastrow, Lemma 2.10diff]. Hence we get an induced homomorphismH�(M) �! H�(N). It is clear that this assignment is functorial.

63

3 Measure Homology

Theorem (3.36) (Long Exact Pair Sequence). There is an exact pair sequence formeasure homology, i.e., if M is a smooth manifold without boundary and N � Mis a smooth submanifold without boundary, then the sequence

�� Hk+1 (M, N) Hk (N)Hk(i)

Hk (M)Hk(j)

Hk (M, N) ��

is exact for all k 2 N. Here i : N ,! M and j : (M, ;) ,! (M, N) are the in-clusions. The so-called connecting homomorphism H�+1(M, N) �! H�(N) isnatural in M and N .

Proof. The definition of relative measure chain groups yields a short exactsequence

0 Ck (N) Ck (M) Ck (M, N) 0 ,

where the maps are induced by the (smooth) inclusions i and j. Thereforewe obtain the desired long exact homology sequence [Weibel, Theorem 1.31].

Theorem (3.37) (Homotopy Invariance). Measure homology is a homotopy in-variant functor, i.e., if f , g : M �! N are homotopic smooth maps of smooth man-ifolds without boundary, then

H�(f) = H�(g) : H�(M) �! H�(N).

A proof is given in [Zastrow, 5.3diff].

Remark (3.38). One might suspect that it is necessary for f and g in theabove theorem to be smoothly homotopic. But smooth homotopic maps arealways smoothly homotopic [Lee2, Proposition 10.22]. �

Using a barycentric subdivision process, Zastrow proves the following the-orem [Zastrow, Theorem 4.1diff]:

Theorem (3.39) (Excision). Measure homology satisfies the following variation ofthe excision axiom: Let M be a smooth manifold without boundary and let W � Mbe a smooth submanifold without boundary. If U � W is a subset such that MnU isa smooth manifold without boundary and W nU � M nU is a smooth submanifoldwithout boundary and if there is some subset V � M such that

U �Æ

V � V � W ,

then the inclusion (M nU, W nU) ,! (M, W) induces an isomorphism

H�(M nU, W nU) �= H�(M, W).

64


Definition (3.40). A topological space X is called normal if disjoint closedsets A, B � X can be separated by open sets, i.e., there are disjoint open setsU, V � X with A � U and B � V . �

Remark (3.41). All metric spaces are normal. In particular, every smoothmanifold is normal (since every smooth manifold can be given a Riemannianstructure, and thus yielding a metric). �

Corollary (3.42). Let M be a smooth manifold without boundary and W � M

an open subset. If U � M is a closed subset with U � W , then the inclusion(M nU, W nU) ,! (M, W) induces an isomorphism

H�(M nU, W nU) �= H�(M, W).

Proof. Since W � M is open, W is a smooth submanifold without boundary.Moreover, M n U is a smooth manifold without boundary and the subsetW n U � M n U is a smooth submanifold without boundary because U isclosed.

Thus it remains to show that there is an open subset V � M with

U � V � V � W .

Since M is normal and U � W , we can separate the disjoint closed sets U

and M nW by open sets V and V 0. In particular,

U � V � V � M nV 0� W .

Hence excision proves the corollary.

Corollary (3.43) (Mayer-Vietoris Sequence). Suppose M is a smooth manifoldwithout boundary and U, V � M are open subsets with

U [V = M.

Then the sequence

�� Hk+1 (U \V ) A Hk+1 (U) �Hk+1 (V ) B Hk+1 (M) � Hk (U \V ) ��

is exact for all k 2 N, where

jU : U \V ,! U, iU : U ,! M,jV : U \V ,! V , iV : V ,! M

65

3 Measure Homology

are the inclusion maps,

A :=�Hk+1 (jU),�Hk+1 (jV)

�,

B :=�Hk+1 (iU),Hk+1 (iV)

�,

and � is defined below.

Proof. By Corollary (3.42) the inclusion (V , U \ V ) ,! (M, U) induces anisomorphism H�(V , U \V ) �= H�(M, U) . We define � by the composition

Hk+1 (M) Hk+1 (M, U) Hk+1 (V , U \V )�= Hk (U \V )

where the arrow on the left is induced by the inclusion M ,! (M, U) andthe arrow on the right is the connecting homomorphism of the long exactmeasure homology sequence of the pair (V , U \V ).

The long exact measure homology sequences of the pairs (V , U \ V ) and(M, U) yield a commutative ladder

��Hk+1 (U \V ) Hk+1 (V ) Hk+1 (V , U \V )�=

Hk (U \V ) ��

�� Hk+1 (U) Hk+1 (M)

�

Hk+1 (M, U) Hk (U) ��

with exact rows. Thus the sequence in the theorem is exact [tom Dieck,Lemma (8.3) in Kapitel IV].

Moreover, measure homology is compatible with colimits.

Theorem (3.44). Let M be a smooth manifold without boundary covered by anascending family (Uk)k2N of open subsets. Then the inclusions Uk ,! M inducean isomorphism

colimk!1

H�(Uk) �= H�(M) .

The idea of the proof is essentially the same as in singular homology: to usethe compactness of the standard simplex. However, we have to be carefulabout some technicalities. The key observation is the following:

Lemma (3.45). Let M and (Uk)k2N be as in the theorem. Suppose j 2 N and� 2 Cj (M) . Then we can find an index k 2 N and a measure chain � 2 Cj (Uk)such that

� = �jk and @� = (@�)jk,

66


where jk : map1 (�j, Uk) �! map1 (�

j, M) is the map induced by the inclu-sion Uk ,! M. Furthermore, if � is a cycle, then so is �.

Proof (of Lemma (3.45)). By Corollary (3.26) the family (im jk)k2N is an openascending covering of map1 (�

j, M), using the notation

8k2N im jk := jk�map1 (�

j, Uk)�.

Let D be a compact determination set for �. Hence there is an index k 2 Nsuch that D � jk

�map1 (�

j, Uk)�.

Let Ak be the Borel �-algebra of im jk (with the subspace topology). ByLemma (3.25) the set im jk is open in map1 (�

j, M). Thus Ak is containedin the Borel �-algebra of map1 (�

j, M). Therefore we may define the signedmeasure e� on im jk via

8A2Ake�(A) := �(A).

By construction, D � im jk, implying that D is a compact determination setof e� . Using Lemma (3.4), it is easy to see that e� has finite total variation.

If Jk : im jk �! map1 (�j, Uk) is the inverse of jk, then Jk is continuous by

Lemma (3.25). Hence the push-forward

� := e�Jk

is a signed measure on map1 (�j, Uk) having a compact determination set

and finite total variation (Lemma (3.9)), i.e., � 2 Cj (Uk).

Our next step is to show the equality � = �jk: Let A � map1 (�j, M) be some

Borel set. HenceeA := A \ im jk 2 Ak

and�(A) = �( eA) + �

�A \ (map1 (�

j, M) n im jk)�.

Since map1 (�j, M) n im jk is a measurable set contained in the complement

of the determination set D,

�(A) = �( eA)

holds. Therefore the constructions of jk and Jk yield

�(A) = �( eA) = e�( eA)

= e��J�1

k

�j�1k

( eA)��

= e�Jk�j�1k

(A \ im jk)�

= ��j�1k

(A \ im jk)�= �

�j�1k

(A)�

= �jk(A).

67

3 Measure Homology

Since jk is induced by the (smooth) inclusion Uk ,! M, the map �jk is a chainmap. In particular,

@� = @(�jk) = (@�)jk.

Suppose � is a cycle and A � map1 (�j, Uk) is a Borel set. Then jk(A) =

J�1k

(A) lies in Ak. As mentioned above, this results in jk(A) being Borelin map1 (�

j, M). Using the injectivity of jk we therefore compute

@�(A) = @��j�1k

(jk(A))�= (@�)jk

�jk(A)

�= @�

�jk(A)

�= 0.

Hence � is also a cycle.

Proof (of Theorem (3.44)). For each k 2 N and each ` 2 N�k we write

ik : Uk ,! M and i`k : Uk ,! U`

for the inclusions. Let j 2 N. Then�Hj (Uk)

�k2N

becomes a direct systemwith respect to the structure maps

�Hj(i

`k)�(k,`)2D where

D := f(k, `) 2 N2j` � kg.

Hence we can form the direct limit colimk!1 Hj (Uk) of this direct system.

The universal property of the direct limit provides us with a homomorphism

hj : colimk!1

Hj (Uk) �! Hj (M)

induced by the maps�Hj (ik)

�k2N

. To see why hj happens to be an isomor-phism we use the following fact [Bredon, Corollary D.3]: it is sufficient toshow that for each � 2 Hj (M) there exists a k 2 N and a � 2 Hj (Uk) suchthat Hj (ik)� = �, and that for each � 2 Hj (Uk) with Hj (ik)(�) = 0 there isan ` 2 N such that Hj

�i`k�(�) = 0.

1. Let � 2 Cj (M) be a measure cycle. Then the preceding lemma providesus with some k 2 N and a measure cycle � 2 Cj (Uk) such that � = �jk.This proves

Hj (ik)�[�]

�= [�]

in Hj (M) , since jk is induced by the inclusion ik.

2. Let k 2 N and � 2 Cj (Uk) be a measure cycle such that Hj (ik)�[�]

�=

0 2 Hj (M) . Hence there exists a measure chain � 2 Cj+1 (M) with

@� = �jk.

68


The above lemma gives us an ` 2 N (and the proof shows that we canassume ` � k) and a measure chain % 2 Cj+1 (U` ) satisfying

%j`= � ,

(@%)j`= @� = �jk.

Writing j`k : map1 (�j, Uk) �! map1 (�

j, U` ) for the map induced bythe inclusion i`k, we conclude

(@%)j` = �jk =��j`

k�j` .

As in the last part of the proof of Lemma (3.45), we can compute for allBorel sets A � map1 (�

j, Uk)

@%(A)= @%�j�1` (j` (A))

�= (@%)j`

�j` (A)

�

=��j`

k�j`�j` (A)

�= �j`

k�j�1` (j` (A))

�

= �j`k(A).

This shows Hj

�i`k��[�]

�= 0 in Hj (U` ).

With the help of these algebraic properties we will see in Section 4.2 thatmeasure homology and singular homology (algebraically) coincide.

69

3 Measure Homology

70

4 Measure Homology isSingular Homology

In this chapter, singular homology (with real coefficients) and measure homologyof smooth manifolds without boundary are compared. The results from Section 3.3already suggest that the homology groups of these two theories should coincide.Surprisingly, the natural isomorphism providing this correspondence turns out tobe isometric. It is thus possible to calculate the simplicial volume of an orientedsmooth closed connected manifold via measure homology.

In order to obtain the mentioned isometric isomorphism between singular homol-ogy and measure homology, we proceed in three steps:

� First of all, we introduce smooth singular homology (for smooth manifoldswithout boundary) and show that it is canonically isometrically isomorphicto singular homology.

� In the second step, we establish an (algebraic) isomorphism between smoothsingular homology and measure homology. This is done by means of theproperties described in Section 3.3.

� Thirdly, we prove that this isomorphism is actually isometric. This is thecentral contribution of this thesis. The method of proof is inspired by theduality principle of bounded cohomology (Theorem (2.7)).

In the last section, we discuss an integration operation for measure chains as a toolto analyse the top homology group, which will turn out to be useful in the contextof smearing (Section 5.4).

71

4 Measure Homology is Singular Homology

4.1 � Smooth singular homology

Since measure homology is based on smooth simplices, it is easier to seea connection between measure homology and smooth singular homologyinstead of singular homology. Smooth singular homology is defined justlike ordinary singular homology, using smooth singular chains instead ofcontinuous ones.

Definition (4.1). Let M be a smooth manifold without boundary and k 2 N.The elements of map1 (�

k, M) are called smooth singular k-simplices of M.The subcomplex (using the inclusion from Remark (3.17))

�Csm� (M,R) , @jCsm

� (M,R)��C�(M,R) , @

�

generated by all smooth singular simplices yields the smooth singular ho-mology groups

Hsmk (M,R) :=

ker�@ : Csm

k (M,R) �! Csmk�1 (M,R)

�

im�@ : Csm

k+1 (M,R) �! Csmk

(M,R)�

for each k 2 N. �

More details about smooth singular homology can be found in the text-books of Massey and Lee [Massey, x2 in Appendix A], [Lee2, Chapter 16]. InMassey’s book it is shown (using the same methods as in ordinary singularhomology) that smooth singular homology satisfies theorems correspond-ing to those stated in Section 3.3. These statements will in the following bereferred to via numbers such as (3.39)sm instead of (3.39).

Moreover, by the same pattern as for singular homology, we can define anorm k�ksm

1 on the smooth singular chain complex and its induced semi-norm k�ksm

1 on smooth singular homology.

Theorem (4.2). For every smooth manifold M without boundary the natural chaincomplex inclusion

jM : Csm� (M,R) ,! C�(M,R)

induces an isometric isomorphism between the homology groups H sm� (M,R) and

H�(M,R) .

72

4.1 Smooth singular homology

Proof. Via the Whitney approximation theorem, a smoothing operator

s : C�(M,R) �! Csm� (M,R)

can be constructed [Lee2, page 417], satisfying the following conditions:

� The map s is a chain map with

sÆ jM = id and jM Æ s ' id .

� For each singular simplex � 2 Sk (M) the image s(�) 2 Csmk

(M,R)consists of just one smooth simplex.

The first part implies that H�(jM ) : Hsm� (M,R) �! H�(M,R) is an isomor-

phism. Its inverse is H�(s).

From the second part we deduce: if P�2Sk(M) a�� 2 Ck (M,R) is a cyclerepresenting the homology class � 2 Hk (M,R) , then P�2Sk(M) a�� s(�) is acycle representing Hk (s)(�) with

kHk (s)(�)ksm1 �

X

�2Sk(M)

a�� s(�)

sm

1� X

�2Sk(M)

ja�j

=

X

�2Sk(M)

a��

1.

Taking the infimum over all representatives of � thus yields

kHk (s)(�)ksm1 � k�k1 =

Hk (jM )

�Hk (s)(�)

�

1 .

On the other hand, obviously Hk (jM )(�0)

1 � �0

sm

1

holds for all �0 2 Hsmk (M,R) . Since Hk (s) is a bijection this proves the

theorem.

In particular, we detect the fundamental class in smooth singular homologyand we can compute the simplicial volume via this class:

Definition (4.3). Let M be an oriented smooth closed connected manifold ofdimension n. The above theorem implies that

[M] sm :=�Hn(jM )

��1�[M]R

�

is a well-defined generator of Hsmn (M,R) . �

73


Corollary (4.4). Let M be an oriented smooth closed connected manifold. Then [M] sm

sm

1 = kMk

holds by the above theorem. �

In Section 4.3, we will make use of a version of bounded cohomology basedon smooth singular homology. Its construction is completely analogous tothe corresponding definition of bounded (singular) cohomology and it turnsout to be the same as ordinary bounded cohomology. However, boundedsmooth cohomology is more feasible in the context of measure homology.

Definition (4.5). Let M be a smooth manifold without boundary, and k 2 N.

� The supremum norm of f 2 HomR

�Csm

k(M,R) ,R

�is given by

kfk 1 := sup�2map1 (�

k,M)

jf(�)j.

� We write

bCksm (M) :=

�f 2 HomR

�Csm

k (M,R) ,R��kfk 1 < 1

for the group of bounded smooth k-cochains.

� The corresponding coboundary operator is defined via

Æ : bCksm (M)�! bCk+1

sm (M)

f 7�!�c 7! (�1)k+1 � f(@(c))

�.

� The k-th bounded smooth cohomology group bHksm (M) of M is the

k-th cohomology group of the cochain complex (bC�sm (M) , Æ).

� The supremum norm induces (as in the case of ordinary bounded co-homology) a seminorm on the quotient bHk

sm (M) , which will also bedenoted by k�k1 . �

Remark (4.6). As in the case of ordinary bounded cohomology, it is easy toverify that the homomorphism Æ is well-defined and satisfies Æ ÆÆ = 0.

Using the smoothing operator s of the proof of Theorem (4.2), it can readilybe seen that bH�

sm (M) and bH� (M) are isometrically isomorphic:

74

4.1 Smooth singular homology

Theorem (4.7). Let M be a smooth manifold without boundary. Then the restric-tion map (where jM is defined as in Theorem (4.2))

bC�(jM ) : bC�(M) �! bC�sm (M)

f 7�! f jCsm� (M,R) = f Æ jM

induces an isometric isomorphism bH� (jM ) : bH� (M) �! bH�sm (M) on cohomol-

ogy.

Proof. It is clear that the restriction map is a well-defined cochain map.

Let h�: C�(M,R) �! C�+1(M,R) be a chain homotopy jM Æ s ' id. Thesame arguments as in the proof of Lemma (2.5) yield that the “dual”

bC�(M) �! bC��1 (M)

f 7�! (�1)�� f Æ h��1

is well-defined and a cochain homotopy bC�(s) Æ bC�(jM ) ' id, where bC�(s)

and bC�(jM ) are the “duals” of s and jM respectively. Moreover,

bC�(jM )Æ bC�(s) = id,

by functoriality. Hence the restriction map induces an isomorphism on co-homology.

Since s maps singular simplices to singular simplices, we obtain

kf Æ sk1 � kfk 1

for all f 2 bC�sm (M) . On the other hand, obviously kf Æ jMk1 � kfk 1 holds

for all f 2 bC�(M) . Therefore we conclude

8'2 bH�(M) k'k 1 =

bH� (s)

�bH� (jM )(')

� 1� bH� (jM )(')

1� k'k 1 ,

which implies that bH� (jM ) is isometric.

As in the case of ordinary bounded cohomology, we can define a Kroneckerproduct

h� ,� i: bC�sm (M) �Csm

� (M,R) �! R.

by evaluation. Analogously, this product descends to a bilinear map

h� ,� i: bH�sm (M) �Hsm

� (M,R) �! R,

75


and it is readily seen to be compatible with the ordinary Kronecker product,i.e., for all ' 2 bH� (M) and all � 2 Hsm

� (M,R)bH� (jM )('), �

�=', H�(jM )(�)

�.

We now obtain a smooth version of the duality principle (compare Theo-rem (2.7)), which will become useful in Section 4.3.

Corollary (4.8) (Duality Principle of Bounded Smooth Cohomology). LetM be a smooth manifold without boundary and let k 2 N and � 2 Hsm

k (M,R) .

1. Then k�ksm1 = 0 if and only if

8'2 bHk

sm (M) h', �i = 0.

2. If k�ksm1 > 0, then

k�ksm1 = sup

n 1k'k 1

��' 2 bHk

sm (M) , h', �i = 1o

.

Proof. By the above remark,

h', �i =bHk (jM )Æ bHk (jM )�1 ('), �

�

=bHk (jM )�1 ('), Hk(jM )(�)

�

holds for all � 2 Hsmk (M,R) and all ' 2 bHk (M) . Since bHk(jM ) and Hk(jM )

are isometric isomorphisms by Theorem (4.7) and Theorem (4.2), the dualityprinciple (2.7) applied to Hk(jM )(�) 2 Hk (M,R) proves the corollary.

4.2 � The algebraic isomorphism

In this section, smooth singular homology (with real coefficients) and mea-sure homology are shown to be isomorphic as vector spaces. This isomor-phism is induced by a natural chain map:

Lemma (4.9). Let M be a smooth manifold without boundary. Then there is anorm-preserving injective chain map

iM : Csm� (M,R) �! C�(M) .

76

4.2 The algebraic isomorphism

Proof. Let k 2 N. For � 2 map1 (�k, M) we denote the atomic measure

on map1 (�k, M) concentrated in � by Æ�, which is obviously an element

of Ck (M) . If c := P�2map1 (�k,M) a�� is a smooth chain, we define

iM (c) := X�2map1 (�

k,M)

a��Æ� 2 Ck (M) .

The calculation showing that iM is a chain map is quite straightforward[Ratcliffe, Lemma 2 in x11.5].

It remains to show that iM is norm-preserving. With the above notation weobtain that the Hahn decomposition of iM (c) is given by

P :=�

� 2 map1 (�k, M)

��a� � 0

,

N :=�

� 2 map1 (�k, M)

��a� < 0

.

Hence we conclude

kiM (c)k = jiM (c)j�map1 (�

k, M)�

= iM (c)(P) � iM (c)(N)

= X�2P

a�� X�2N

a�

= X�2map1 (�

k,M)

ja�j

= kcksm1 .

Theorem (4.10). If M is a smooth manifold without boundary, then the inclusioniM : Csm

� (M,R) ,! C�(M) of the previous lemma induces an isomorphism

Hsm� (M,R) �= H�(M) .

In the case of measure homology based on the compact open topology (in-stead of the C1-topology), proofs of this theorem were given by Hansen andZastrow via the verification of the Eilenberg-Steenrod axioms [Hansen, The-orem 1.1], [Zastrow, Theorem 3.4]. In the smooth setting however, it is notentirely clear how to use this approach since only smooth manifolds can behandled (instead of the whole universe of CW-complexes).

Therefore we give a proof based on some “manifold induction” – followinga general idea of Milnor:

Theorem (4.11). Assume P is some property of smooth manifolds without bound-ary satisfying the following conditions:

77


1. The property P is stable under smooth homotopy equivalences.

2. The property P holds for the one point space (which can be thought of as azero-dimensional smooth manifold without boundary).

3. If U, V are open subsets of the same smooth manifold without boundary suchthat U, V and U \ V have property P , then the union U [ V also has prop-erty P .

4. If (Uk)k2N is an ascending sequence of subsets of some (common) smoothmanifold without boundary all having property P , then

Sk2N Uk satisfies P .

Then each smooth manifold without boundary has property P .

Proof. The (not very difficult) proof can for example be found in Massey’sbook [Massey, Case 5 on page 364].

Proof (of Theorem (4.10)). We consider the property P which is defined as fol-lows: a smooth manifold U without boundary satisfies P if and only if thehomomorphism

H�(iU) : Hsm� (U,R) �! H�(U)

is an isomorphism.

In view of Theorem (4.11) it suffices to show that P has the four mentionedproperties:

1. Suppose f : M �! N is a smooth homotopy equivalence of smooth man-ifolds without boundary where M enjoys P . Then the diagram

Hsm� (M,R)

Hsm� (f ,R)

H�(iM )

Hsm� (N ,R)

H�(iN )

H�(M)H�(f)

H�(N)

is commutative and all arrows but the right vertical one are isomorphisms.Hence H�(iN ) must also be an isomorphism, showing that N fulfils P .

2. The chain complexes C�(�)and Csm� (�,R)are identical and the inclusion

i�: C�(�),! Csm� (�,R)is the identity. Hence the one point space satisfies P .

78

4.2 The algebraic isomorphism

3. Let U, V be open subsets of some smooth manifold M without boundarysuch that U, V , and U \V satisfy P . Then we obtain the following commu-tative ladder

Hsmk+1 (U \V ,R)

Hk+1(iU\V ) �=

Hsmk+1 (U,R)

�H

smk+1 (V ,R)

Hk+1(iu)

�Hk+1(iV)�=

Hsmk+1 (U [V ,R)

Hk+1(iU[V )

Hsmk

(U \V ,R)

Hk(iU\V ) �=

Hsmk

(U,R)�

Hsmk

(V ,R)

Hk(iU)�Hk(iV) �=

Hk+1 (U \V )Hk+1 (U)

�Hk+1 (V )

Hk+1 (U [V ) Hk(U \V )Hk(U)

�Hk(V )

whose rows are Mayer-Vietoris sequences and thus exact (compare Corol-lary (3.43) and (3.43)sm). Now the five lemma implies that the union U [Vhas property P .

4. Let M be a smooth manifold without boundary and (Uk)k2N an ascendingsequence of open subsets of M. Suppose that each Uk satisfies P .

SinceS

k2N UK is a smooth manifold without boundary covered by the as-cending family (Uk)k2N of open sets, we conclude from Theorem (3.44) andTheorem (3.44)sm that the inclusions induce isomorphisms

i : colimk!1

H�(Uk) �! H�(U),

j : colimk!1

Hsm� (Uk,R) �! Hsm

� (U,R)

using the notationU :=

[

k2N

Uk.

Furthermore, for each k 2 N the diagram

Hsm� (Uk,R)

Hsm� (Uk,!U,R)

H�(iUk)

Hsm� (U,R)

H�(iU)

H�(Uk)H�(Uk,!U)

H�(U)

commutes. Since each Uk has property P , we conclude that the inducedhomomorphism

colimk!1

H�(iUk) : colim

k!1Hsm� (Uk,R) �! colim

k!1H�(Uk)

79


is an isomorphism. The previous diagram shows that

colimk!1 Hsm� (Uk,R)

j

colimk!1 H�(iUk)

Hsm� (U,R)

H�(iU )

colimk!1 H�(Uk) iH�(U)

is commutative, implying that H�(iU) is also an isomorphism.

In particular, we obtain a fundamental class for measure homology. It isthe aim of this chapter to show that this fundamental class can be used tocompute the simplicial volume.

Definition (4.12). Let M be a smooth oriented closed connected manifold ofdimension n. By Theorem (4.10) the measure homology fundamental class

[M]mh := Hn(iM )�[M] sm

�2 Hn (M)

is a generator of

Hn (M) �= Hsmn (M,R) �= Hn (M,R) �= R. �

4.3 � . . . is isometric

In this section it will be shown that the (algebraic) isomorphism from Theo-rem (4.10) is actually isometric. Thurston already stated that this should betrue [Thurston, page 6.6], however the paper cited there has never been pub-lished in the suggested form. In early 2004, a preprint by Bowen appearedto remedy this situation [Bowen], but the given argument does not seem tobe correct (see Subsection 4.3.4).

We recall the following notation which will be used throughout this section:

Definition (4.13). If M is a smooth manifold without boundary, we writejM : Csm

� (M,R) ,! C�(M,R) for the obvious inclusion of chain complexesand iM : Csm

� (M,R) �! C�(M) for the injective norm-preserving chain mapwhich is given by mapping smooth singular simplices to the correspondingatomic measure (compare (4.9)). �

80

4.3 . . . is isometric

Then the result can be formulated as follows:

Theorem (4.14). If M is a smooth manifold without boundary, then the inclusioniM : Csm

� (M,R) ,! C�(M) induces an isometric isomorphism

Hsm� (M,R) �= H�(M) .

Before giving the proof we state the main consequence of this theorem:

Corollary (4.15). Let M be a smooth manifold without boundary.

1. Then the inclusions iM and jM induce an isometric isomorphism

H�(M,R) �= H�(M) .

2. In particular, the simplicial volume of M can be computed via measure ho-mology, i.e.,

kMk = [M]mh

mh.

Remark (4.16). The second part was already known to be true for hyper-bolic manifolds [Gromov2, page 235], [Ratcliffe, Exercise 10 in x11.5] but thegeneral case was still open. �

Remark (4.17). It seems possible to use the same proof in the case of mea-sure homology for nice enough metric spaces (based on the compact opentopology on the set of singular simplices), hence giving an isometric iso-morphism between singular homology and measure homology for general(metric) spaces. �

The proof of the theorem is based on the idea of imitating the duality resultsof Theorem (2.7) (and Corollary (4.8)). This yields a reformulation of theproblem in terms of bounded cohomology, which can be solved by means ofthe established framework sketched in Section 2.3.

The drawback of this approach is that the proof of Theorem (4.14) is not asgeometric as one might wish.

4.3.1 � A dual for measure homology

As a first step we have to construct the “dual” bH� (M) which plays the roleof the bounded (smooth) cohomology groups bH�

sm (M) in the (smooth) sin-gular theory. If c = P�2map1 (�

k,M) a�� 2 Csmk

(M,R) is a smooth singular

81


chain and f 2 bCksm (M) is a singular cochain, their Kronecker product is

given byhf , ci = f(c) = X

�2map1 (�k,M)

a�� f(�) 2 R.

If we think of c as a linear combination of atomic measures, this looks like anintegration of f over c. Hence our “dual” in measure homology will consistof (bounded) functions which can be integrated over measure chains:

Definition (4.18). Let M be a smooth manifold without boundary and letk 2 N. We write

bCk(M) :=�

f : map1 (�k, M) �! R

��f is Borel measurable and bounded

and

Æ : bCk(M)�! bCk+1(M)

f 7�!�� 7! (�1)k+1 � f(@(�))

�.

Here f(@(�)) is an abbreviation for Pk+1j=0 (�1)j � f(� Æ@j). �

The next lemma shows that this map Æ : bCk(M) �! bCk+1(M) is indeed well-defined and turns bC�(M) into a cochain complex.

Lemma (4.19). Let M be a smooth manifold without boundary, k 2 N, and letf 2 bCk(M) . Then Æ(f) 2 bCk+1(M) and

Æ ÆÆ(f) = 0.

Proof. According to Remark (3.24),

map1 (�k+1, M) �! map1 (�

k, M)

�7�! � Æ@j

is continuous for each j 2 f0, : : : , k+ 1g. Hence Æ(f) is Borel measurable. Asf is bounded, Æ(f) is bounded as well.

Since @(@(�)) = 0 holds for all � 2 map1 (�k+2, M), it follows that

Æ ÆÆ(f) = 0.

Definition (4.20). Let M be a smooth manifold without boundary and letk 2 N. The k-th bounded measure cohomology group of M is given by

bHk (M) := Hk�bC�(M) , Æ

�.

We write k�k1 for the seminorm on bHk (M) which is induced by the supre-mum norm on bCk(M) . �

82


Definition (4.21). Let M be a smooth manifold without boundary and letk 2 N. The Kronecker product of � 2 Ck (M) and f 2 bCk(M) is defined as

hf , �i :=

Z

f d�.

If � is a measure cycle and f is a cocycle, we write

[f ], [�]

�:= hf , �i =

Z

f d�. �

It is easy to see that this is well-defined and – as desired – yields a generali-sation of the ordinary Kronecker product:

Lemma (4.22). Let M be a smooth manifold without boundary and let k 2 N.

1. The Kronecker product h� ,� i: bCk(M) �Ck (M) �! R is well-defined andbilinear.

2. The Kronecker product h� ,� i: bHk (M) �Hk (M) �! R on (co)homology iswell-defined and bilinear.

3. The Kronecker product defined above is compatible with the Kronecker producton bounded smooth cohomology in the following sense: for all f 2 bCk(M) andall c 2 Csm

k (M,R) ,hB(f), ci = hf , iM (c)i ,

where B(f) denotes the linear extension of f : map1 (�k, M) �! R to the

vector space Csmk

(M,R) . Passage to (co)homology yields for all ' 2 bHk (M)and all � 2 Hsm

k(M,R)

Hk(B)('), �

�=', Hk(iM )(�)

�.

Proof. 1. The integral is defined and finite, since the elements of Ck (M) are(signed) measures of finite total variation and the elements of bCk(M) arebounded measurable functions. Moreover, the integral is obviously bilinear.

2. It remains to show that the definition of the Kronecker product on (co)ho-mology does not depend on the chosen representatives. This is a conse-quence of the following fact: if ` 2 N, � 2 C` (M) and f 2 bC`�1 (M) , thenthe transformation formula yields

Z

map1 (�` ,M)f Æ@j d� =

Z

map1 (�`�1 ,M)f d�(�7!�Æ@j)

83


for all j 2 f0, : : : , `g, and henceR

Æ(f) d� =R

f d@(�).

3. Since both Kronecker products are bilinear, it suffices to consider the casewhere c consists of a single smooth simplex � 2 map1 (�

k, M). Then the lefthand side – by definition – evaluates to f(�). Since iM (c) = iM (�) = Æ� is theatomic measure on map1 (�

k, M) concentrated in � , we obtain for the righthand side

hf , iM (c)i =

Z

f dÆ�= 1� f(�).

The corresponding equality in (co)homology follows because B : bC�(M) �!bC�

sm (M) is easily recognised to be a chain map.

The above Kronecker product leads to the following (slightly weakened) du-ality principle.

Lemma (4.23) (Duality Principle of Measure Homology). Let M be a smoothmanifold without boundary, k 2 N, and � 2 Hk (M) .

1. If k�kmh = 0, then for all ' 2 bHk (M)

h', �i = 0.

2. If k�kmh > 0, then

k�kmh � supn 1k'k 1

��' 2 bHk (M) , h', �i = 1

o.

Proof. Let ' 2 bCk(M) . Assume that � 2 Ck (M) is a measure cycle represent-ing � and f 2 bCk(M) is a cocycle representing '. If h', �i = 1, then

1 = jh', �i j = jhf , �i j

=��

Z

f d��

Z

jf j d�+ +

Z

jf j d��

� kfk 1 �

Z

1 d�+ + kfk 1 �

Z

1 d��

= kfk 1 �k�k .

Taking the infimum over all representatives results in 1 � k'k 1 �k�kmh . Inparticular, if there exists such a ', then

k�kmh �1

k'k 1

> 0.

Now the lemma is an easy consequence of this inequality.

84


A posteriori we will be able conclude – in view of Theorem (4.14), andLemma (4.25) – that in the first part of the lemma “if and only if” is alsotrue and that in the second part equality holds.

4.3.2 � Proof of Theorem (4.14)

The dual bC�(M) will be investigated by means of the complex I�(M) intro-duced in Definition (2.36): the vector space Ik (M) is the set of all boundedfunctions in map( eMk+1 ,R) which are �1(M)-invariant. Then the key to theproof of Theorem (4.14) is a careful analysis of the diamond

I�(M)

DA

bC�(M) ,

E

bC�(M)

B

bC�sm (M)

the maps being defined as follows:

� For k 2 N let sk : map(�k, M) �! map(�k, eM) be a Borel section of themap induced by the universal covering map. The existence of such asection is guaranteed by the following theorem:

Theorem (4.24). Let M be a smooth manifold without boundary with uni-versal covering map p : eM �! M, and let k 2 N.

1. Then the map

P : map��k, eM

��! map

��k, M

�

�7�! pÆ�

is a local homeomorphism.

2. Moreover, there exists a Borel section of P .

The (elementary, but rather technical) proof of this theorem is exiled toSubsection 4.3.3.

85


If f 2 Ik (M) , we write

A(f) : map1��k, M

��! R

�7�! f�(sk(�))(e0), : : : , (sk(�))(ek)

�,

using the inclusion map1 (�k, M) ,! map(�k, M).

� Similarly (see also Section 2.5), we define D(f) for f 2 Ik (M) as thelinear extension to Ck (M,R) of the map

Sk (M) = map��k, M

��! R

�7�! f�(sk(�))(e0), : : : , (sk(�))(ek)

�.

� The map B is given by linear extension (cf. Lemma (4.22)).

� The map E is given by restriction, i.e., E = bC�(jM ).

This diagram allows us to compare the map B (which is the building bridgebetween bH� (M) and bH�

sm (M) ) with D and E, which both induce isometricisomorphisms on the level of bounded cohomology.

Lemma (4.25). With the above notation the following statements hold:

1. The maps A, B, D, and E are well-defined cochain maps.

2. Moreover, kA(f)k 1 � kfk 1 for all f 2 Ik (M) and all k 2 N. This implies

8�2Hk(I�(M) )

H�(A)(�)

1� k�k1 .

3. The diagram commutes, that is B ÆA = E ÆD.

4. The homomorphism H�(B) : bH� (M) �! bH�sm (M) induced by the cochain

map B is surjective.

Proof. 1. The fact that B and E are well-defined and compatible with therespective coboundary operators can be directly read off the definitions.

In Section 2.5 it is proved that D does not depend on the choice of sk andthat D is a cochain map. The same calculations show that the definitionof A is independent of the choice of sk and that A is a cochain map. Let

86


f 2 Ik (M) . Then A(f) is obviously bounded. Moreover, A(f) is measurable:By construction, A(f) is the composition

map1��k, M

�,! map

��k, M

� sk�! map��k, eM

�! eMk+1 f

�! R

�7! � � 7! (�(e0), : : : , �(ek))

which consists of measurable and continuous functions (see Lemma (3.23),Remark (3.24), Remark (3.11)).

2. This is an immediate consequence of the definition of A.

3. For all f 2 Ik (M) and all � 2 map1 (�k, M) we obtain

�B ÆA(f)

�(�)=

�A(f)

�(�)

= f�(sk(�))(e0), : : : , (sk(�))(ek)

�

=�D(f)

��jM (�)

�

=�E ÆD(f)

�(�).

4. Due to Corollary (2.38) and Theorem (4.7), the chain maps D and E induceisomorphisms on cohomology. Since the above diagram commutes, H�(B)must be surjective.

We have now collected all the necessary tools to prove that measure homol-ogy and (smooth) singular homology are isometrically isomorphic.

Proof (of Theorem (4.14)). According to Theorem (4.10), the induced homo-morphism H�(iM ) : Hsm

� (M,R) �! H�(M) is an isomorphism. Therefore,it remains to show that H�(iM ) is compatible with the seminorms.

Let k 2 N and � 2 Hsmk

(M,R) . Since iM : Csm� (M,R) �! C�(M) is norm

preserving, it is immediate that

kHk(iM )(�)kmh � k�ksm1 .

The proof of the reverse inequality is split into two cases:

➀ Suppose Hk(iM )(�)

mh = 0. From Lemma (4.22) and Lemma (4.23) weobtain

Hk(B)('), ��=', Hk(iM )(�)

�= 0

for all ' 2 bHk (M) . By the previous lemma, bHk (B) is surjective. Hence

8 2 bHk

sm (M) h , �i = 0,

87


implying k�ksm1 = 0.

➁ Let Hk(iM )(�)

mh > 0. In this case, Lemma (4.23) and Lemma (4.22)yield

Hk(iM)(�)

mh � supn 1k'k 1

�� ' 2 bHk (M) ,

', Hk(iM)(�)

�= 1

o

= supn 1k'k 1

��' 2 bHk (M) ,

Hk(B)('), �

�= 1

o.

We will compare the last set with the corresponding set of Corollary (4.8):Let 2 bHk

sm (M) such that h , �i = 1. Since the composition Hk(E)ÆHk(D)is an isometric isomorphism (Corollary (2.38) and Theorem (4.7)), there ex-ists a � 2 Hk

�I�(M)

�satisfying

�Hk(E)ÆHk(D)

�(�) = and k�k1 = k k 1 .

Then ' := Hk(A)(�) 2 bHk (M) possesses the following properties:

� By construction,

Hk(B)(') =�Hk(B)ÆHk(A)

�(�) =

�Hk(E)ÆHk(D)

�(�) = ,

and henceHk(B)('), �

�= h , �i = 1.

� Furthermore, we get from Lemma (4.25)

k'k 1 = Hk(A)(�)

1� k�k1 = k k 1 .

Combining these properties with the above estimate results in

Hk(iM )(�)

mh � supn 1k'k 1

��' 2 bHk (M) ,

Hk(B)('), �

�= 1

o

� supn 1k k 1

�� 2 bHk

sm (M) , h , �i = 1o

.

Since k�ksm1 � kHk(iM )(�)kmh > 0, we can use Lemma (4.8) to conclude

kHk(iM)(�)kmh � k�ksm1 .

88


4.3.3 � Existence of a Borel section

To complete the proof of Theorem (4.14), we still have to provide a proof ofTheorem (4.24):

Proof (of Theorem (4.24)). Since M is smooth, we may assume that eM is alsoa smooth manifold and p is a local diffeomorphism. Moreover, the smoothmanifold M can be equipped with a Riemannian metric. Then there is aRiemannian metric on eM such that p is a local isometry (e.g., the pulledback Riemannian metric). The topologies on M and eM are induced by themetrics dM and d eM

which are given by the respective Riemannian metrics.Then p is also a local isometry with respect to dM and d eM

.

1. Let � 2 map(�k, eM). Since p : eM �! M is a covering map, there is a smallneighbourhood U of �(e0) in eM , which means:

Definition (4.26). An open subset U � eM is called small if p(U) � M isopen and pjU : U �! p(U) is a homeomorphism. �

We will show that P(Ufe0g ) is open and that P jUfe0g : Ufe0g �! P(Ufe0g ) is a

homeomorphism.

The set P(Ufe0g ) is open in map(�k, M). By definition, P(Ufe0g ) ��p(U)

�fe0g .

On the other hand, for each � 2�p(U)

�fe0g there exists a lift e� : �k �! eM

such that e�(e0) 2 U since �k is simply connected. Thus

P(Ufe0g ) =�p(U)

�fe0g .

Since p(U) is open in M, this is an open subset of map(�k, M).

The restriction P jUfe0g : Ufe0g �!

�p(U)

�fe0g = P�Ufe0g

�is bijective. Since U

is small, pjU is injective. Hence the uniqueness of lifts (prescribed on e0 bythe property to map into U) shows injectivity (�k is connected).

The restriction P jUfe0g : Ufe0g �!

�p(U)

�fe0g = P�Ufe0g

�is a homeomorphism.

Because of Lemma (3.15), the map P is continuous. It therefore remains toprove that the restriction P j

Ufe0g is open:

Let % 2 Ufe0g . Since �k is compact and pÆ% is continuous, the set

K := pÆ%(�k)

is compact.

89


Then the “distance” of the sheets of p over K is uniformly bounded frombelow in the following sense:

Lemma (4.27). There is a constant C 2 R>0 satisfying the following property: thedistance of all points y1, y2 2 p�1 (K) such that p(y1) = p(y2) and y1 6= y2 isbounded from below via

d eM(y1, y2) � C.

Proof (of Lemma (4.27)). We consider the function

D : K �! R

x 7�! infy2p�1 (x)nfexg

d eM(ex, y),

where ex is some p-lift of x. The function D is independent of this choice sincethe fundamental group �1(M) acts isometrically on eM and transitively oneach fibre (Corollary (5.10) and [Massey, Lemma V 8.1]).

Let x 2 K. Then there is an open neighbourhood W of x such that there is atrivialisation

W � �1(M)'W�=

prW

p�1 (W)

p

W

and pj'W(W�f1g) : 'W(W � f1g) �! W is an isometric diffeomorphism.

Let ex := 'W(x, 1). Since W is an open neighbourhood of x, the image'W(W �f1g) is an open neighbourhood of ex in p�1 (W) (the fibre �1(M) car-ries the discrete topology) and hence in eM . So there is an rx 2 R>0 such thatthe open ball B

eM2rx

(ex) is contained in 'W(W � f1g) and p�B

eM2rx

(ex)�= BM

2rx(x).

Let z be a point in the open neighbourhood Vx := BMrx (x) of x. We have

BMrx(x) � W because pj'W(W�f1g) is isometric and BM

2rx(x) = p

�B

eM2rx

(ex)�.

Since 'W is injective and BeM

2rx(ex) � 'W(W � f1g), we conclude

'W(z, h) 62 BeM

2rx(ex)

for all h 2 �1(M) n f1g. In particular,

d eM

�'W(z, h), 'W(z, 1)

�� d eM

�'W(z, h), ex

�� d eM

�ex, 'W(z, 1)

�

� 2 � rx � dM (x, z)

� 2 � rx � rx

= rx

90


for all h 2 �1(M) n f1g. This yields D(z) � rx for all z 2 Vx. Since K iscompact, it can be covered by finitely many such sets Vx1 , : : : , Vxn . Then

C := minj2f1,:::,ng

rxj

gives the desired estimate.

We will show below that for all small enough " 2 R>0 the image of

eV" :=n

� 2 map(�k, eM)�� sup

x2�k

d eM(�(x), %(x)) < "

o

under P jUfe0g is the set

V" :=n

� 2 map(�k, M)�� sup

x2�k

dM (�(x), pÆ%(x)) < "o

.

As �k is compact, the topologies on map(�k, eM) and map(�k, M) are thesame as the respective topologies of uniform convergence (Remark (3.13)).This proves that P j

Ufe0g is open.

The idea is to apply the above lemma in the following way: since the sheetsof the covering are far away from each other and functions in V" are veryclose, their lifts must lie in the “same” sheet. Using the fact that p is a localisometry, we can conclude that the lifts are also very close.

In order to make the second statement precise, we need the following result:

Lemma (4.28). There is a constant Æ 2 R>0 such that for each y 2 p�1 (K) therestriction

pjB

eMÆ(y)

: BeM

Æ (y) �! p�B

eMÆ (y)

�

to the open ball of radius Æ around y (with respect to d eM) is an isometric diffeomor-

phism and p�B

eMÆ (y)

�= BM

Æ (p(y)).

Proof (of Lemma (4.28)). For each x 2 K we choose an element ex 2 p�1 (x).Then there are balls B

eM"x(ex) such that pj

BeM"x(ex)

: BeM

"x(ex) �! p

�B

eM"x(ex)

�is an

isometric diffeomorphism and p�B

eM"x(ex)

�= BM

"x(x).

Since �1(M) acts by isometries on eM (see Corollary (5.10)), we conclude that

g �BeM

"x(ex) = B

eM"x(g� ex)

91


and thatpj

g�BeM"x(ex)

: g �BeM

"x (ex) �! p�B

eM"x (ex)

�

is an isometric diffeomorphism for all g 2 �1(M). As K is compact, we canfind an n 2 N and x1, : : : , xn 2 K such that

K �[

j2f1,:::,ng

BM"xj/2(xj).

Then

Æ := minj2f1,:::,ng

"xj

2

is the desired constant: namely, let y 2 p�1 (K) and x := p(y). Then thereis a j 2 f1, : : : , ng such that x 2 BM

"xj/2(xj). Since �1(M) acts transitively on

each fibre, there is a g 2 �1(M) satisfying y 2 g�BeM

"xj(exj). Because pj

BeM"xj

(g� exj)is an isometry,

d eM(y, g� exj) = dM (x, xj) �

"xj

2.

In particular, BeM

Æ (y) � BeM

"xj(g� exj). But then

pjB

eMÆ(y)

: BeM

Æ (y) �! p�B

eMÆ (y)

�

is also an isometric diffeomorphism. Moreover,

BMÆ (x) � BM

"xj(xj) = p

�B

eM"xj(exj)

�,

since x 2 BM"xj/2(xj). Therefore p

�B

eMÆ (y)

�= BM

Æ (p(y)).

How small does " have to be? We take " 2�0, min(C/4, Æ/3)

�so small that

eV"� Ufe0g , where C is the constant of Lemma (4.27) and Æ is constructed inLemma (4.28).

Since " < Æ, we conclude that P( eV") � V": Let � 2 eV" and x 2 �k. Since%(x) 2 p�1 (K) and " < Æ, it follows that

dM

�pÆ �(x), pÆ%(x)

�= d eM

��(x), %(x)

�< ".

Hence P(�) = pÆ � 2 V".

92


Conversely, let � 2 V". Since �k is simply connected, there is a lift e� of � .Moreover, we may choose e�(e0) in such a way that

d eM

�e�(e0), %(e0)

�< "

because %(e0) 2 p�1 (K) and dM

��(e0), pÆ %(e0)

�< " < Æ. In order to show

that e� 2 eV"we have to check that the set

D :=�

x 2 �k��d eM

�e�(x), %(x)

�< "

equals �k.

By construction, e0 2 D. In particular, D is not empty. Since �k is connected,it suffices to prove that D is open and closed:

� The set D is open. Let x 2 D. Since e� and % are continuous, there is anopen neighbourhood W of x such that

8y2W d eM

�e�(y), e�(x)

�< " and d eM

�%(y), %(x)

�< ".

Hence

d eM

�e�(y), %(y)

�� d eM

�e�(y), e�(x)

�+ d eM

�e�(x), %(x)

�+ d eM

�%(x), %(y)

�

< "+ "+ " � Æ

for all y 2 W . Now the construction of Æ yields

d eM

�e�(y), %(y)

�= dM

��(y), pÆ%(y)

�< ".

Thus W � D, implying that D is open.

� The set D is closed. Suppose x 2 �k n D, i.e., d eM(e�(x), %(x)) � ".

The condition " < Æ assures that p is isometric on the "-ball in eM

around %(x) and that p�B

eM" (%(x))

�= BM

" (p Æ %(x)). In particular, theball B

eM" (%(x)) contains a point y such that p(y) = �(x) because � 2 V".

Now Lemma (4.27) yields

d eM

�e�(x), %(x)

�� d eM

�e�(x), y

�� d eM

�y, %(x)

�

� C � ".

By continuity, the set W � �k of all z satisfying d eM(e�(z), e�(x)) < "

and d eM(%(z), %(x)) < " is an open neighbourhood of x. Therefore, we

obtain for all z 2 W

d eM

�e�(z), %(z)

�� d eM

�e�(x), %(x)

�� d eM

�e�(z), e�(x)

�� d eM

�%(x), %(z)

�

� C � "� "� " � ".

Hence W � �k nD, which shows that D is closed.

93


This proves the first part of Theorem (4.24).

2. As for the second part note that we can cover map(�k, eM) with countablymany open sets (Vn)n2N on which P is a homeomorphism (e.g., one couldtake a countable covering of eM by small sets U and consider sets of theform Ufe0g ). In particular, the sets (P(Vn))n2N are open in map(�k, M). ViaW0 := P(V0) and

8n2N Wn+1 := P(Vn+1) n[

j2f0,:::,ng

Wj

we get a countable family (Wn)n2N of mutually disjoint Borel subsets inmap(�k, M) such that P�1 jWn

is well-defined and continuous for each n 2 N.Moreover, map(�k, M) is covered by the (Wn)n2N because P is surjective.Putting all these maps together yields the desired Borel section of P .

4.3.4 � Bowen’s argument

In the following, flaws in Bowen’s preprint “An Isometry Between MeasureHomology and Singular Homology” [Bowen] are outlined, showing that hisapproach does not work. In this section, we will use the notation of hispreprint.

Bowen’s idea is to first improve measure cycles (without changing the rep-resented class in homology) and then explicitly construct a singular chainrepresenting the same class. Both steps are designed in such a manner thatthe norm is not increased.

For the first step a map R : C�(M) �! C�(M) is constructed, mapping mea-sure chains to these improved chains. Bowen claims that this map R is chainhomotopic to the identity. However, it is easy to see that R is not even achain map in general.

Furthermore, the inductive construction of simplices for the second step isnot possible: e.g., let � be the 2-simplex wrapping around S2 and mappingthe boundary @�2 to the point x0 2 S2. Let � be the 2-simplex which mapseverything to x0 (in particular, �j@�2 = � j@�2). Then � ' � relative to thevertices of �2, but not � ' � relative to the boundary @�2.

This example also shows that, in general, [j(c)] 6= [�]: Let � := j(� � �). Thisis a cycle and � generates H2

�S2,R

��= R. However, � 2 H(�). Therefore,

j(c) = ��H(�)

�� j(�) = (1 � 1)� j(�) = 0.

In particular, [j(c)] 6= [�].

94

4.4 Integrating measure homology chains

4.4 � Integrating measure homology chains

As in (smooth) singular homology theory (Lemma (1.10)), we would like tointroduce an integration process for measure chains to get information aboutthe measure homology classes represented by certain cycles.

Definition (4.29). Let M be a smooth manifold without boundary. Let k 2N, � 2 map1 (�

k, M) and ! be a smooth k-form on M. Then there is anextension (f , V ) of � and we define

Z

�

! :=

Z

�k

f�!,

where f�! denotes the pulled back k-form on the k-manifold V . �

Integrating forms over smooth simplices is indeed well-defined (i.e., inde-pendent of the chosen extension) since the differentials of all extensions coin-cide on T�k (Lemma (3.20)). Hence the pulled back forms coincide over �k,implying that the integral is well-defined.

Remark (4.30). If M is an oriented Riemannian manifold without boundaryof dimension n, we get for all � 2 map1 (�

n+1, M) via the generalised StokesTheorem [Lee2, Theorem 16.10]

Z

@�

volM =

Z

�

d volM

(where volM is the volume form on M). Since d volM is an (n + 1)-form ofthe n-dimensional manifold M, it must be zero. Hence

Z

@�

volM = 0.

Thus the linear extension Csmn (M,R) �! R of integrating the volume form

over a smooth singular simplex induces a homomorphism

Hsmn (M,R) �! R

[c] 7�!

Z

c

volM

on homology. �

95


The following lemma reveals the reason for using the C1-topology on themapping space map1 (�

k, M) instead of the coarser compact open topology.The map described in the Lemma would not be continuous when workingwith the compact open topology (since the pulled back form considerablydepends on the differential).

Lemma (4.31). Let M be a smooth manifold without boundary, k 2 N, and let !

be some smooth k-form on M. Then the map

map1��k, M

��! R

�7�!

Z

�

!

is continuous.

Proof. This is an immediate consequence of the definition of the C1-topologyand the compatibility of integration with certain limits [Ratcliffe, Lemma 3in x11.5].

Corollary (4.32). Let M be a smooth manifold without boundary, k 2 N, and let! be a smooth k-form on M. For each � 2 Ck (M) the map

: map1��k, M

��! R

�7�!

Z

�

!

is �-integrable.

Proof. Let D be a compact determination set of �. By the previous lemma, is continuous. In particular, is Borel measurable and the restriction jD isbounded.

The mapping space map(�k, M) is Hausdorff since M is Hausdorff. Thusmap1 (�

k, M) is Hausdorff by Lemma (3.23). In particular, the compact set D

is closed and hence Borel. This implies that

:= �D �

is a bounded measurable function (where �D stands for the characteristicfunction of the Borel set D). Moreover, map1 (�

k, M) n D must be a �-nullset and hence also a null set with respect to �+ and �� .

96

4.4 Integrating measure homology chains

Since � has finite total variation, Lemma (3.4) implies

�+(D) = �(D \P) < 1,

�� (D) = ��(D \N) < 1,

where map1 (�k, M) = P [ N is a Hahn decomposition for �. Therefore

the characteristic function �D is �+-integrable and �� -integrable. Since is almost everywhere (with respect to both �+ and �� ) dominated by theintegrable function �D �maxx2D (x), it is itself integrable with respect to �+

and �� .

Hence we can integrate forms over measure chains:

Definition (4.33). Let M be a smooth manifold without boundary, k 2 N,and let ! be some smooth k-form on M. For each measure chain � 2 Ck (M)we define Z

�

! :=

Z

map1 (�k,M)

Z

�

! d�(�). �

In fact, this integration is a generalisation of integration over smooth sim-plices in the following sense:

Lemma (4.34). Let M be a smooth manifold without boundary, k 2 N, and let !

be some smooth k-form on M. For each smooth chain c 2 Csmk (M,R)

Z

iM (c)! =

Z

c

!

holds, iM : Csm� (M,R) ,! C�(M) being the inclusion described in Lemma (4.9).

Proof. Since integration is linear, we only need to consider smooth singularsimplices � 2 Csm

k(M,R) . By definition of iM ,Z

iM (�)! =

Z

map1 (�k,M)

Z

�

! d�iM (�)

�(�)

=

Z

�

! � 1,

which proves the lemma.

The above compatibility statement and Lemma (1.10) can be combined toyield the following description of the top measure homology group:

97


Theorem (4.35). Let M be an oriented closed connected Riemannian manifold ofdimension n, and let � 2 Cn (M) be a measure cycle. Then the measure homologyclass [�] can be expressed as follows:

[�] =

R� volM

vol(M)� [M]mh 2 Hn (M) .

Proof. The transformation formula shows [Ratcliffe, Lemma 4 in x11.5]

8�2Cn+1(M)

Z

@�

volM =

Z

�

d volM =

Z

�

0 = 0.

Thus integration of the volume form induces a homomorphism

Hn (M) �! R

[�]7�!

Z

�

volM .

Denoting the canonical inclusion Csmn (M,R) ,! Cn (M) by iM , we obtain the

commutative diagram (compare Lemma (4.34))

Hsmn (M,R)

R

�volM

Hn(iM )

R.

Hn (M)

R

�volM

The vertical arrow is an isomorphism by Theorem (4.10) mapping the fun-damental class [M] sm to the measure homology fundamental class [M]mh.

According to Lemma (1.10)Z

c

volM = vol(M)

holds, whenever c 2 Csmn (M,R) is a representative of [M] sm. HenceZ

�

volM =

Z

c

volM = vol(M)

for each measure cycle � representing the measure homology fundamentalclass [M]mh.

98

5 The ProportionalityPrinciple

The proportionality principle of simplicial volume reveals a fascinating connectionbetween the simplicial volume and the Riemannian volume: simplicial volume andRiemannian volume are proportional if we restrict our attention to manifolds whichshare the same universal Riemannian covering space. Similar proportionality prin-ciples also occur in the setting of L2-invariants (Theorem (1.41)) and in the settingof characteristic numbers of complex manifolds [Hirzebruch, Satz 3].

Both Thurston and Gromov sketched (dual) proofs for the proportionality principleof simplicial volume [Thurston, page 6.9], [Gromov2, Section 2.3]. Thurston’s idea isto use measure homology to compute the simplicial volume and to take advantageof the larger chain complex for the so-called smearing construction. In this chapter,a detailed version of Thurston’s proof will be given, based on the results of theprevious chapters.

Gromov’s proof uses arguments from bounded cohomology and also depends onsome averaging operation via Haar measures. Unfortunately his exposition is notvery explicit about certain measurability issues.

In Section 5.1, the proportionality principle is discussed and the strategy of proof ispresented. Sections 5.2 and 5.3 provide the necessary tools for Thurston’s smearingtechnique (i.e., the study of isometry groups and appropriate measures on them). InSection 5.4, the smearing of smooth singular chains is developed. The actual proofof the proportionality principle is given in Section 5.5. The chapter ends with someapplications in Section 5.6.

99

5 The Proportionality Principle

5.1 � Statement of the proportionality principle

The proportionality principle of simplicial volume reveals a fascinating con-nection between the simplicial volume and the Riemannian volume:

Theorem (5.1) (Proportionality Principle of Simplicial Volume). Let M andN be oriented closed connected Riemannian manifolds with isometrically isomor-phic universal Riemannian coverings. Then

kMk

vol(M)=

kNk

vol(N).

When speaking of a universal Riemannian covering map, we mean a uni-versal covering map which is an orientation preserving local isometry.

Without loss of generality, we may assume that the universal Riemanniancovering spaces of M and N are orientation preservingly isometrically isomor-phic. Otherwise we just reverse the orientation of M.

Thus it is clear that we even may assume that M and N have the same univer-sal Riemannian covering since the composition of an orientation preservingisometry and a universal Riemannian covering map again is a universal Rie-mannian covering map.

For convenience, we introduce the following notation, which will be used inlater sections.

Setup (5.2). Let M and N be two oriented closed connected Riemannianmanifolds with common universal Riemannian covering U. The coveringmaps U �! M and U �! N will be denoted by pM and pN respectively.Furthermore, we write n for the dimension of M and N . �

In order to prove the proportionality principle, we are looking for a chainmap C�(N ,R) �! C�(M,R) inducing multiplication by vol(N)/ vol(M)on the top homology group without increasing the seminorm on homology.Then the inequality

vol(N)

vol(M)�kMk � kNk

100

5.1 Statement of the proportionality principle

would easily follow. Swapping the roles of M and N would give the reverseinequality and hence prove the proportionality principle.

Integration (for smooth simplices) shows that the naive approach

� 7�! pM Æ e�

(e�being some pN -lift of �) would yield the factor vol(N)/ vol(M) if it werea chain map (which it is not).

So we may try to apply some averaging operation (“smearing”) to elimi-nate this choice of lift to get a chain map. But since �1(N) is not finite ingeneral, there is no direct way to do this. Using measure homology chainsinstead of singular chains, we can take advantage of the Haar measure onthe locally compact group Isom+

1 (U) of orientation preserving isometrieson U, which contains �1(N). However, to satisfy the finiteness conditionfor measure chains, the compact quotient �1(M) n Isom+

1 (U) is more appro-priate. To guarantee the norm condition, the constructed measure on thequotient �1(M) n Isom+

1 (U) is scaled to a probability measure.

The resulting chain map

smearN ,M : Csm� (N ,R) �! C�(M)

indeed does not increase the seminorm and induces the desired factor on thelevel of homology – as can be seen by integration.

The smearing technique was invented by Thurston [Thurston, page 6.8].Thurston used it primarily to prove Theorem (1.23), which he calls “Gro-mov’s Theorem” (and which is by Gromov described as “Thurston’s Theo-rem”). Thurston also sketches a proof of the proportionality principle basedon smearing [Thurston, page 6.9]. However, he uses the isometric isomor-phism of Theorem (4.14), but gives no evidence for it (in fact, a paper is citedwhich never appeared in the suggested form). A more detailed discussionof smearing is given in Ratcliffe’s book [Ratcliffe, x11.5].

The proportionality constant in the proportionality principle is only knownin very few special cases such as hyperbolic manifolds (Theorem (1.23)) orflat manifolds (Corollary (5.26)).

101


5.2 � Isometry groups

As indicated in Section 5.1, the principal underlying tool for smearing isthe measure on a quotient of the group of orientation preserving isometrieson the universal covering manifold induced by the Haar measure on thisisometry group. So we first examine this isometry group.

Whenever we refer to a Riemannian manifold as a metric space, it is under-stood that the metric is the one induced by the Riemannian metric.

Remark (5.3). Let (U, g) be a Riemannian manifold without boundary. Amap f : U �! U is a Riemannian isometry (i.e., f is smooth and the dif-ferential Tf transforms the Riemannian metric g into g) if and only if it is ametric isometry [Helgason, Theorem I 11.1]. �

Definition (5.4). Let U be a Riemannian manifold (without boundary, butnot necessarily compact).

� The group of (metric) isometries on U, written as Isomco(U), is the setof all metric isometries U �! U with the multiplication given by com-position, endowed with the subspace topology induced by the com-pact open topology on map (U, U).

� If U is oriented, the group of orientation preserving isometries on U,denoted by Isom+

1 (U), is the mapping space consisting of all orien-tation preserving isometries U �! U with the multiplication givenby composition. The topology on Isom+

1 (U) is the subspace topologyinduced by the C1-topology on map1 (U, U). �

From the topologist’s point of view it is of course much more comfortable towork with the group Isomco(U). However, we are forced to use the finerC1-topology on the space of smooth singular simplices to make the inte-gration process in Lemma (4.31) continuous. Therefore – as can be seen inLemma (5.16) – we also have to take the C1-topology on the isometry group.We will see a posteriori (Theorem (5.12)) that the compact open topologyand the C1-topology on the isometry groups coincide. But there does notseem to exist a short way to prove this fact.

102

5.2 Isometry groups

Before working out basic properties of Isom+1 (U), we collect some facts

about the coarser group Isomco(U). Lemma (5.7) will be the key for applyingthese results to Isom+

1 (U).

Theorem (5.5). Let U be a Riemannian manifold.

1. The group Isomco(U) is indeed a topological group with respect to the giventopology.

2. The set Isomco(U) is closed in map (U, U).

3. The group Isomco(U) is Hausdorff, locally compact, and satisfies the secondcountability axiom.

4. If G is a group acting properly and cocompactly on U by isometries, then G

is a discrete and cocompact subgroup of Isomco(U).

Proof. 1. A proof is given by, e.g., Helgason [Helgason, Theorem IV 2.5].

2. If f 2 map (U, U) n Isomco(U), there are x, y 2 U such that

d�f(x), f(y)

�6= d(x, y),

where d denotes the metric on U determined by the Riemannian metric. Inparticular, x 6= y. Since d : U � U �! R�0 is continuous, there are openneighbourhoods V and W of f(x) and f(y) respectively such that

8v2V 8w2W d(v, w) 6= d(x, y).

Furthermore, fxg and fyg are obviously compact. Hence V fxg \W fyg is anopen neighbourhood of f in map (U, U) satisfying

�V fxg \W fyg

�\ Isomco(U) = ;,

by construction of V and W .

Parts 3. and 4. follow since Riemannian manifolds are second countableproper metric spaces [Sauer, Theorem 2.35]. The cited proof is based onthe Arzela-Ascoli Theorem, and for this to apply, it is crucial that Isomco(U)is closed in map (U, U).

Since the actual object of interest is the group Isom+1 (U), we would like to

have a similar theorem for Isom+1 (U). This will be accomplished by equip-

ping the tangential space of U with a Riemannian structure in a canonicalway:

103


Remark (5.6). Let U be a Riemannian manifold. On the tangent bundle TU,a Riemannian metric can be defined as follows [do Carmo, Chapter 3, Exer-cise 2]:

Let (p, q) 2 TU and v, w 2 T(p,q)(TU). Choose smooth curves

�v = (pv, qv) : [�1, 1 ] �! TU,�w = (pw, qw) : [�1, 1 ] �! TU

representing v and w respectively, i.e., pv(0) = p = pw(0), qv(0) = q = qw(0)and v = �0v(0), w = �0w(0). Then the inner product of v and w is given by

hv, wi(p,q) :=T(p,q)�(v), T(p,q)�(w)

�p+ hD0qv, D0qwip

where � : TU �! U is the bundle projection, D0qv denotes the covariantderivative of the vector field qv along pv (with respect to the Riemannianconnection, of course), and analogously for D0qw. �

Lemma (5.7). Let U be an oriented Riemannian manifold and f 2 Isom+1 (U).

ThenTf 2 Isomco(TU).

Proof. The construction of the smooth structure on the tangent bundle TU

shows that Tf : TU �! TU is smooth. Therefore it remains to show that Tfpreserves the Riemannian metric on TU:

Suppose (p, q) 2 TU and v, w 2 T(p,q)(TU). Choose smooth curves

(pv, qv) : [�1, 1 ] �! TU,(pw, qw) : [�1, 1 ] �! TU

representing v and w respectively. Using the chain rule and the fact that� ÆTf = f Æ �, we conclude

T(f(p),Tpf(q)) ��T(p,q)Tf(v)

�= Tpf

�T(p,q)�(v)

�.

Since f is an isometry of the Riemannian manifold U, this yieldsT(f(p),Tpf(q)) �

�T(p,q)Tf(v)

�, T(f(p),Tpf(q)) �

�T(p,q)Tf(w)

��f(p)

=Tpf

�T(p,q)�(v)

�, Tpf

�T(p,q)�(w)

��f(p)

= hT(p,q)�(v), T(p,q)�(w)ip.

104

5.2 Isometry groups

Considering the second summand in the definition of hv, wi(p,q), we observethat (by naturality of the Riemannian connection under isometries [Lee1,Proposition (5.6)])

D0(Tf Æ qv) = Tf(D0qv)

which implies (since f is an isometry)D0(Tf Æ qv), D0(Tf Æ qw)

�f(p)

=Tf(D0qv), Tf(D0qw)

�f(p)

= hD0qv, D0qwip.

Thus the Riemannian metric on TU is preserved by the map Tf , i.e., Tf isan isometry of TU.

In view of Lemma (5.7) we can use the results and methods of Theorem (5.5)to derive a corresponding theorem for the group Isom+

1 (U):

Theorem (5.8). Let U be an oriented Riemannian manifold.

1. The group Isom+1 (U) is indeed a topological group with respect to the topol-

ogy from Definition (5.4).

2. The group Isom+1 (U) is Hausdorff and satisfies the second countability ax-

iom.

3. The set T�Isom+

1 (U)�

is closed in map1 (TU, TU).

4. Moreover, the group Isom+1 (U) is locally compact.

5. If G is a group acting properly and cocompactly on U by isometries, then G

is a discrete and cocompact subgroup of Isom+1 (U).

Proof. Lemma (5.7) shows that T�Isom+

1 (U)�� Isomco(TU). Therefore

T : Isom+1 (U) �! T

�Isom+

1 (U)�� Isomco(TU)

is (by definition of the topology on Isom+1 (U)) a homeomorphism.

1. Since multiplication and inverting elements on Isomco(TU) are contin-uous according to Theorem (5.5), this is also true for the restrictions to thesubset T

�Isom+

1 (U)�. Thus Isom+

1 (U) is a topological group.

2. Since Isomco(TU) is Hausdorff by Theorem (5.5), the same is true for thesubset T

�Isom+

1 (U)�. Therefore Isom+

1 (U) is also Hausdorff. By the sameargument, Isom+

1 (U) is second countable.

105


3. Using the fact that Isomco(TU) is Hausdorff and second countable, wecan check the closedness of T

�Isom+

1 (U)�

via sequences:

Let (gn)n2N � T�Isom+

1 (U)�

be a sequence converging to g 2 Isomco(TU).Hence we can write

8n2N gn = Tfn

for an appropriate sequence (fn)n2N � Isom+1 (U). The convergence of the

sequence (Tfn)n2N to g implies that (fn)n2N converges to

g := �U Æ g Æ iU

in the compact open topology (where iU : U �! TU is the inclusion ofthe zero-section and �U : TU �! U denotes the bundle projection). SinceiU and �U are smooth, it follows that g is smooth. Theorem (5.5) impliesthus that g is an isometry.

To convince ourselves that Tg = g, we use (special) charts of TU to reducethe problem to a similar one in Euclidean space (compare Lemma (5.9)),where it can easily be solved.

Let (V1, '1) be a relatively compact chart around x 2 U (i.e., the closure of V1is compact) and let (V2, '2) be a chart around g(x) 2 U such that g(V1) � V2.Since each of the gn = Tfn preserves fibres, the same must be true for theirlimit g. Then

g��1

U (V1)�� 1

U (V2).

By definition of the manifold structure on TU, the pairs��1

U (V1), T'1�

and��1

U (V2), T'2�

are charts of TU around each point in ��1U (x) and ��1

U (g(x))respectively.

Since V1 is relatively compact and V2 is open, fn(V1) � V2 holds for all largeenough n 2 N. So, without loss of generality, we may assume that this istrue for all n 2 N. This implies

Tfn��1

U (V1)�� 1

U (V2)

for all n 2 N since Tfn is fibre-preserving. Thus we may define

G := T'2 Æ gÆ (T'1)�1 ,

Fn := T'2 ÆTfn Æ (T'1)�1 .

106

5.2 Isometry groups

In particular, we get commutative diagrams of the form (with d := dim(U))

��1U (V1)

T'1

gn j=Tfnj��1

U (V2)

T'2

'1(V1)�Rd Fn

'2(V2)�Rd.

On the other hand, the differential Tfn is characterised by the fact that theabove diagram commutes when the map on the bottom is substituted by

'1(V1)�Rd �! '2(V2)�R

d

(y, z) 7�!�('2 Æ fn Æ '�1

1 )(y), J('2 Æ fn Æ '�11 )y(z)

�

(here J stands for the Jacobian matrix). Hence Fn must be this (smooth) map(the vertical arrows are diffeomorphisms). Moreover, (Fn)n2N convergesto G in the compact open topology since T'1 and T'2 are homeomorphisms.

Thus we can apply the following lemma (whose proof is deferred until theend of the proof of the theorem):

Lemma (5.9). Let d 2 N, let V1, V2 � Rd be two open sets, and (fn)n2N �map1 (V1, V2). Assume that the maps (Fn)n2N � map1

�V1 �R

d, V2 �Rd�,

defined by

Fn : V1 �Rd�! V2 �R

d

(y, z) 7�!�fn(y), (Jfn)y(z)

�

converge in the compact open topology to a function G : V1 � Rd �! V2 � R

d.Then G1 : V1 �! V2 is continuously differentiable and

8(y,z)2V1�Rd (JG1)y(z) = G2(y, z),

where (denoting the obvious inclusions/projections by i1, p1, p2)

G1 := p1 ÆG Æ i1,G2 := p2 ÆG.

Since the image of the zero-section is closed in TU and preserved by all thegn = Tfn, the same must hold for g. Then a straightforward computationshows

'2 Æ g Æ ('1)�1 = p1 ÆG Æ i1 =: G1.

107


Therefore the lemma shows

8(y,z)2'1(V1)�Rd(Jg)y(z) = p2 ÆG(y, z) =: G2(y, z),

yielding the commutative diagram

��1U (V1)

T'1

gj��1

U (V2)

T'2

'1(V1)�Rd

(G1,JG1)'2(V2)�R

d.

On the other hand, we already know that g is smooth. So the above diagramproves

gj��1U(V1)

= Tgj��1U(V1)

.

Moreover, the map g is orientation preserving: A local diffeomorphism be-tween oriented smooth manifolds is orientation preserving if and only if itsJacobian matrix with respect to any oriented smooth charts has positive de-terminant [Lee2, Exercise 13.5]. Using the same argument as above, we getthat the Jacobians of the (fn)n2N converge pointwise to the Jacobian of g

with respect to such charts. Since the determinant is continuous and each fnpreserves the orientation, their limit g must also be orientation preserving.

4. To see that Isom+1 (U) is locally compact, we apply Theorem (5.5) to the

group Isomco(TU). Thus for each map f 2 Isom+1 (U) the differential Tf 2

Isomco(TU) possesses a compact neighbourhood V . Then V \T�Isom+

1 (U)�

is a compact neighbourhood of Tf in T�Isom+

1 (U)�

because T�Isom+

1 (U)�

is a closed subset of the Hausdorff space Isomco(TU). Therefore

T�1�V \T�Isom+

1 (U)��

is a compact neighbourhood of f in Isom+1 (U).

5. Due to Theorem (5.5), we know that G is a discrete subgroup of Isomco(U).Since the topology on Isom+

1 (U) is finer than the compact open topology, itis clear that G � Isom+

1 (U) is discrete.

We now proceed as in Sauer’s proof of Theorem (5.5) [Sauer, Theorem 2.35]:Choose a point x 2 U. Since G acts cocompactly on U, we can find a compactset K � U such that [Elstrodt, Lemma VIII 3.19]

G �K = M.

108

5.2 Isometry groups

Hence for every isometry f 2 Isom+1 (U) there exists an element gf 2 G such

that gf Æ f(x) 2 K. We consider the set

KG :=�

T(gf Æ f)�� f 2 Isom+

1 (U)

.

According to Lemma (5.7), we know that all the elements of KG have Lip-schitz constant 1. Moreover,

KG(0x) :=�

T(gf Æ f)(0x)�� f 2 Isom+

1 (U)

��

0y

�� y 2 ��1 (K)

= i(K),

where � : TU �! U denotes the bundle projection, 0y (for y 2 U) is the zeroelement of the tangent space TyU and i : U �! TU is the zero section. ThusKG(0x) is relatively compact.

Therefore, we can apply the Arzela-Ascoli Theorem [Sauer, Lemma 2.13],which shows that KG is relatively compact in map (TU, TU). Since the im-age T

�Isom+

1 (U)�

is a closed subset of the Hausdorff space map1 (TU, TU),it follows that KG \ T

�Isom+

1 (U)�

is relatively compact in T�Isom+

1 (U)�.

HenceT j�1

Isom+1 (U)

(KG) = fgf Æ f j f 2 Isom+1 (U)g

must be relatively compact in Isom+1 (U).

Denoting the canonical projection Isom+1 (U) �! G n Isom+

1 (U) by �G, weget by construction that the restriction

�GjT j�1Isom+

1 (U)(KG)

: T j�1Isom+

1 (U)(KG) �! G n Isom+

1 (U)

is surjective, implying the compactness of the quotient G n Isom+1 (U).

To complete the proof of the previous theorem, it remains show Lemma (5.9):

Proof (of Lemma (5.9)). Convergence in the compact open topology coincideswith uniform convergence on compact subsets (Remark (3.13)). In particu-lar, the fn converge pointwise to G1 for n ! 1.

Since each Fn preserves fibres and is linear in each fibre, the same holds fortheir limit G. If K � V1 is compact, the product K � Sd�1 is a compact subsetof V1 � R

d. Hence the Fn converge uniformly on K � Sd�1 to G for n ! 1.This implies that

Jfn �! G2

109


uniformly on K for n ! 1 (with respect to the operator norm).

A standard theorem from calculus [Lee2, Theorem A.70] now shows that G1is continuously differentiable and that

8(y,z)2V1�Rd (JG1)y(z) = G2(y, z).

Corollary (5.10). Let M be an oriented Riemannian manifold without boundaryand U its Riemannian universal covering. Then �1(M) can be considered a sub-group of Isom+

1 (U) and the quotient �1(M) n Isom+1 (U) is compact.

Proof. Covering theory shows that �1(M) acts on U from the right by (mu-tually distinct) homeomorphisms. Since the covering map �M : U �! M isan orientation preserving local isometry and

U��

�M

U

�M

Mid

M

is commutative for each � 2 �1(M), it follows that �1(M) acts on U byorientation preserving isometries. Moreover, multiplication in �1(M) cor-responds to composition in Isom+

1 (U) [Massey, Corollary V 7.5]. Hence�1(M) is a subgroup of Isom+

1 (U).

The quotient �1(M) n Isom+1 (U) is compact: By covering theory, �1(M) acts

properly on U [Massey, page 136] and �1(M) nU �= M is compact [Massey,Lemma 8.1 in Chapter V]. Thus �1(M) n Isom+

1 (U) is compact by the lastpart of Theorem (5.8).

Using the results of Theorem (5.8), we will now be able to conclude that thecompact open and the C1-topology on the isometry groups coincide (but wewill not need this fact in the sequel). Unfortunately, I was not able to figureout a proof of this fact which avoids the nasty computations in the proof ofTheorem (5.8).

Definition (5.11). For a Riemannian manifold U, we write Isom1 (U)for thegroup of isometries endowed with the subspace topology induced by theC1-topology on map1 (U, U). �

Theorem (5.12). Let U be a Riemannian manifold. Then the topologies on thegroups Isomco(U) and Isom1 (U) are the same.

110

5.3 The Haar measure

Proof. Exactly the same methods as in the proof of Theorem (5.8) show thatIsom1 (U) is a topological group which is Hausdorff, locally compact, andsecond countable.

We will now make use of the methods described in Kechris’ book “Clas-sical Descriptive Set Theory” [Kechris]. One of the central notions is thatof a standard Borel space, which provides a convenient setting for measuretheory.

As locally compact second countable Hausdorff groups, both Isomco(U)and Isom1 (U) are standard Borel groups [Kechris, Theorem 5.3]. Sincethe C1-topology is finer than the compact open topology, the identity mapid : Isom1 (U) �! Isomco(U) is a bijective Borel map. Then id alreadyis a Borel isomorphism [Kechris, Theorem 14.12]. Hence Isom1 (U) andIsomco(U) coincide as Borel spaces.

This suffices to conclude that the topologies on Isom1 (U) and Isomco(U)are the same [Kechris, Proposition 12.25].

5.3 � The Haar measure

In the following, some basic properties of Haar measures are recalled. Haarmeasures are regular measures on locally compact groups which are com-patible with the group structure. Using the results from the previous section,we will prove that there is a right invariant probability measure on the Borel�-algebra of the quotient �1(M) n Isom+

1 (U). This measure will be used inthe averaging process mentioned in the introduction.

Definition (5.13). Let G be a locally compact Hausdorff topological group,and let A be its Borel �-algebra.

� A (positive) measure � on A is called regular if

�(A) = sup�

�(K)��K compact, K � A

,

�(A) = inf�

�(U)��U open, A � U

holds for all A 2 A, and if �(K) is finite for all compact sets K � G.

111


� A Haar measure on G is a positive measure h on A which is regular,non-zero on any non-empty open set, and which is left invariant, i.e.,

h(g�A) = h(A)

for all g 2 G and all A 2 A.

� The group G is called unimodular if each left invariant Haar measureon G also is right invariant. �

It is well-known that such a Haar measure exists for each locally compactHausdorff group and is essentially unique (up to multiplication with a pos-itive constant) [Elstrodt, Satz VIII 3.12].

For example, the Haar measure on the discrete group Z is the counting mea-sure, and the Haar measure on the additive group R is the Lebesgue mea-sure. Obviously, all Abelian groups are unimodular. Additionally, all com-pact groups are unimodular [Elstrodt, Satz VIII 3.16], and – as we will show– the isometry group Isom+

1 (U) of an oriented Riemannian manifold is uni-modular.

Definition (5.14). Let X be a topological space, let � be a measure on theBorel �-algebra of X, and suppose that the group G acts via Borel isomor-phisms on X (from the left). A measure fundamental domain of this actionis a Borel set F � X such that

�(X nG �F) = 0,

and8g2Gnf1g �(g�F \F) = 0. �

Theorem (5.15). Presuppose the notation from Setup (5.2). There exists a (pos-itive) right invariant measure hM on the Borel �-algebra of �1(M) n Isom+

1 (U)satisfying

hM

��1(M) n Isom+

1 (U)�= 1.

Proof. Using properties of standard Borel spaces, it can be shown that eachdiscrete cocompact subgroup of a locally compact second countable Haus-dorff group G has finite covolume with respect to the Haar measure (i.e.,there is a measure fundamental domain of finite measure for this subgroup)[Sauer, page 42]. Hence G must be unimodular [Sauer, Lemma 2.32].

112

5.3 The Haar measure

Since Isom+1 (U) is a locally compact second countable Hausdorff topological

group (Theorem (5.8)) and �1(M) � Isom+1 (U) is discrete and acts cocom-

pactly on Isom+1 (U) (Corollary (5.10)), we conclude that �1(M) has finite

covolume in Isom+1 (U) and that Isom+

1 (U) is unimodular.

It follows that the Haar measure h on Isom+1 (U) descends to a right invari-

ant measure h0 on the quotient �1(M) n Isom+1 (U) [Elstrodt, Korollar 3.25

in Kapitel VIII]. If F is a finite measure fundamental domain of �1(M) and�M : Isom+

1 (U) �! �1(M) n Isom+1 (U) is the projection, then one can show

that

h0(A) = h��1

M (A) \F�

holds for all Borel subsets A � �1(M) n Isom+1 (U):

Indeed, for each g 2 �1(M) the translate F � g is also a measure fundamen-tal domain of finite measure because h is right invariant. Left invarianceof h yields that h

��1

M (� ) \ F�

is independent of the chosen fundamentaldomain F . Then it is easy to see that h

��1

M (� ) \ F�

is a right invariant mea-sure on the quotient �1(M) n Isom+

1 (U).

Since �1(M) has finite covolume in Isom+1 (U), it follows that h0

��1(M) n

Isom+1 (U)

�is finite. Why is h0

��1(M) n Isom+

1 (U)�6= 0? Since �1(M) is

countable and F is a measure fundamental domain, we get (where the num-ber of non-zero summands might be infinite)

h�Isom+

1 (U)�= X

g2�1(M)

h(g� F)

= Xg2�1(M)

h(F).

In addition, h�Isom+

1 (U)�6= 0 since Haar measures are by definition non-

trivial. Thus

0 6= h(F) = h0��1(M) n Isom+

1 (U)�.

So

hM :=h0

h0��1(M) n Isom+

1 (U)�

is a measure with the desired properties.

113


5.4 � Smearing

Using the methods of the preceding sections, we are now able to introducethe main construction – the “smearing” of smooth singular chains.

In this section we will always presuppose the notation from Setup (5.2). Asa first step, we define for each smooth simplex � on U a measure smearM (�)which is supported on all pM Æ � , where � is an (orientation preserving) iso-metric translate of � and which is uniformly distributed – the simplex � is“smeared” over the set pM Æ

�Isom+

1 (U)��.

Subsequently, we will pull back this construction to smooth singular sim-plices on N . Using the results from Section 4.4, we will show that smear-ing induces indeed the desired factor vol(N)/ vol(M) on the top homology.Moreover, it will be fundamental that smearing does not increase the norm.

Lemma (5.16). Let k 2 N and � 2 map1 (�k, U). The map

F�: �1(M) n Isom+1 (U) �! map1

��k, M

�

��1(M)

�� f 7�! pM Æ f Æ�

is well-defined and continuous.

Since we take the C1-topology on the right hand side, we are forced to con-sider the C1-topology on the isometry group Isom+

1 (U) to get a continuousmap.

Proof. For all � 2 �1(M) we get

pM Æ (��) = pM

since �1(M) acts by deck transformations, implying that F� is independentof the chosen representative.

By Lemma (3.15) and Remark (3.24) the map

Isom+1 (U) �! map1

��k, M

�

f 7�! pM Æ f Æ�

is continuous. Thus the universal property of the quotient topology yieldsthe continuity of F�.

114

5.4 Smearing

Definition (5.17). For each k 2 N and each � 2 map1 (�k, U), we define the

smeared chain of � bysmearM (�) := hF�

M ,

where hM is the measure from Theorem (5.15). We extend smearM linearlyto the whole chain group Csm

k (U,R). �

By Lemma (5.16), the map F� is, in particular, Borel measurable. HencesmearM (�) is a well-defined measure on map1 (�

k, M).

Lemma (5.18). 1. For each k 2 N and each smooth chain c 2 Csmk (U,R) we

have smearM (c) 2 Ck (M) .

2. The map smearM : Csm� (U,R) �! C�(M) is a chain map.

Proof. 1. Of course, we need only show this assertion for smooth singularsimplices � 2 map1 (�

k, U). According to Lemma (3.4), hM is a signed mea-sure of finite total variation because hM � 0 and

hM

��1(M) n Isom+

1 (U)�= 1.

Furthermore, �1(M)n Isom+1 (U) is a compact determination set of hM . Now

Lemma (3.9) yieldssmearM (�) = hF�

M 2 Ck (M)

since F� is, due to Lemma (5.16), continuous.

2. Let k 2 N, j 2 f0, : : : , k + 1g, and let � 2 map1 (�k+1, U) be a smooth

singular (k+ 1)-simplex. By linearity, it is sufficient to show

smearM (� Æ@j) = @j

�smearM (�)

�

where � Æ@j stands for the j-th face of � . By construction,

smearM (� Æ@j)= hF�Æ@j

M

= h

�(�1(M))�f7!pMÆfÆ�Æ@j

�

M

=�hF�

M

�(�7!�Æ@j)

= @j

�hF�

M

�

= @j

�smearM (�)

�.

To pull back the smearing operation to smooth singular simplices on N viapN -lifts, we need the right invariance of the measure hM to guarantee well-definedness:

115


Lemma (5.19). Let k 2 N and � 2 map1 (�k, N).

1. Then there is a pN -lift of � lying in map1 (�k, U). In fact, each continuous

pN -lift of � is smooth.

2. If �1, �2 2 map1 (�k, U) are two pN -lifts of � , then

hF�1M = h

F�2M .

Proof. 1. Since �k is simply connected, there is a continuous pN -lift

e�: �k �! U

of � . Noting that the covering projection pN : U �! N is a local diffeomor-phism (even a local isometry), we deduce from the commutative diagram

U

pN

�k�

e�

N

that e�also must be smooth (smoothness is a local property).

2. Since �k is connected, two continuous pN -lifts of � only differ by someelement in �1(N). As already mentioned in Corollary (5.10), the fundamen-tal group �1(N) acts by orientation preserving isometries on the universalcover U. Hence there is a g 2 Isom+

1 (U) such that

�1 = gÆ�2.

Writing rg : �1(M) n Isom+1 (U) �! �1(M) n Isom+

1 (U) for right multiplica-tion by g, we get F�1= F�2Æ rg . Thus,

hF�1M = h

F�2ÆrgM =

�h

rgM

�F�2.

Finally, the right invariance of hM enters the scene: Since hM is right invari-ant, we have

hrgM = hM ,

which completes the proof.

Hence we can pull back smearM to smooth singular chains on N :

116

5.4 Smearing

Definition (5.20). For each k 2 N and each smooth singular simplex � 2map1 (�

k, N) we define its smeared chain by

smearN ,M (�) := smearM (e�) 2 Ck (M) ,

where e�2 map1 (�k, U) is some pN -lift of � . Again, the notation smearN ,M

will also be used for its linear extension Csm� (N ,R) �! C�(M) . �

Lemma (5.21). The smearing map

smearN ,M : Csm� (N ,R) �! C�(M)

is a chain map.

Proof. Let k 2 N, j 2 f0, : : : , k + 1g, and let � 2 map1 (�k+1, N). If e� is a

pN -lift, then e�Æ @j is obviously a pN -lift of � Æ @j. Hence we conclude fromLemma (5.18)

smearN ,M (� Æ@j)= smearM (e�Æ@j)

= @j

�smearM (e�)

�

= @j

�smearN ,M (�)

�.

Thus smearN ,M is a chain map.

Furthermore, smearing does not spoil the norm:

Lemma (5.22). For all k 2 N and all c 2 Csmk

(N ,R)

ksmearN ,M (c)k � kcksm1 .

Proof. Since the measure hM is positive, so is smearN ,M (�) for all smoothsingular simplices � 2 map1

��k, N

�. This implies

ksmearN ,M (�)k= smearN ,M (�)�map1 (�

k, U)�

= hM

��1(M) n Isom+

1 (U)�

= 1.

Now the general case is an immediate consequence of the triangle inequalityfor the total variation (compare Lemma (3.7)).

The integration operation introduced in Section 4.4 makes it possible to un-derstand the effect smearN ,M has on homology:

117


Theorem (5.23). Smearing induces multiplication by vol(N)/ vol(M) on the tophomology group. More precisely: for all cycles z 2 Csm

n (N ,R) with [z] = [N]smin Hsm

n (N ,R),

�smearN ,M (z)

�=

vol(N)


The proof basically relies on the following calculation:

Lemma (5.24). For all � 2 map1 (�k, N) the following integrals are equal:

Z

smearN ,M (�)volM =

Z

�

volN .

Proof (of Lemma (5.24)). Let e�be some pN -lift of � . By definition, the integralcan be computed as follows:

Z


Z

map1 (�k,M)

Z

�

volM d�smearN ,M (�)

�(�)

=

Z

map1 (�k,M)

Z

�

volM d�h

Fe�

M

�(�).

Using the transformation formula for measures, we get

Z


Z

�1(M)nIsom+1 (U)

Z

Fe�(�1(M)�f)

volM dhM (�1(M)� f)

=

Z

�1(M)nIsom+1 (U)

Z

pMÆfÆe�

volM dhM (�1(M)� f)

=

Z

�1(M)nIsom+1 (U)

Z

e�

f�p�M volM dhM (�1(M)� f).

By definition, pM and pN are orientation preserving local isometries. Sinceorientation preserving local isometries between Riemannian manifolds pull

118

5.4 Smearing

back the volume form to the volume form, we concludeZ


Z

�1(M)nIsom+1 (U)

Z

e�

volU dhM (�1(M)� f)

=

Z

�1(M)nIsom+1 (U)

Z

e�

p�N volN dhM

=

Z

�1(M)nIsom+1 (U)

dhM �

Z

e�

p�N volN

= 1�Z

pNÆe�

volN

=

Z

�

volN .

Proof (of Theorem (5.23)). Lemma (5.24) and Lemma (1.10) implyZ

smearN ,M (z)volM =

Z

z

volN = vol(N).

Application of Theorem (4.35) therefore results in

�smearN ,M (z)

�=

vol(N)


Remark (5.25). It is possible to extend the smearing chain map smearN ,M toall measure chains:

Ck (N)�! Ck (M)

� 7�!

�

A 7!

Z

map1 (�k,N)h

Fe�

M (A) d�(�)

�

.

This smearing map also does not increase the norm and induces multiplica-tion by vol(N)/ vol(M) on top homology. This leads to an intrinsic proof forthe proportionality principle of the “simplicial volume” defined via measurehomology. �

119


5.5 � Proof of the proportionality principle

Combining the techniques from Section 5.4 and the isometric isomorphismof Section 4.3, we are now in state to do the final step in the proof of theproportionality principle of simplicial volume:

Proof (of Theorem (5.1)). Suppose z 2 Csmn (N ,R) is a cycle which represents

the (smooth) fundamental class [N]sm. Theorem (5.23) shows that smearN ,Minduces the factor vol(N)/ vol(M) on top homology, i.e.,

�smearN ,M (z)

�=

vol(N)

vol(M)� [M]mh .

Since we can compute the simplicial volume via measure homology (Corol-lary (4.15)) and since smearing does not increase the norm (Lemma (5.22)),we obtain

kMk = k[M]mhkmh

�vol(M)

vol(N)�ksmearN ,M (z)k

�vol(M)

vol(N)�kzksm

1 .

Now taking the infimum over all (smooth) representatives of [N]sm gives theestimate

kMk �vol(M)

vol(N)�k[N]smk

sm1 =

vol(M)

vol(N)�kNk

because k[N]smksm1 = kNk by Corollary (4.4). Swapping the roles of the

manifolds M and N shows the reverse inequality

kNk �vol(N)

vol(M)�kMk

and hence proves the proportionality principle.

120

5.6 Applications of the proportionality principle

5.6 � Applications of the proportionality principle

Using Gromov’s upper bound for the simplicial volume in terms of the min-imal volume (Theorem (1.29)), we already have seen that the simplicial vol-ume of flat oriented closed connected manifolds must vanish. We will givenow a proof of this fact based on the proportionality principle instead ofapplying Gromov’s sophisticated estimate:

Corollary (5.26). If M is an oriented closed connected flat Riemannian manifold(i.e., it is locally isometric to Euclidean space), then

kMk = 0.

Proof. For c 2 R>0 we consider the Riemannian manifold Mc which is ob-tained from M by scaling its Riemannian metric by the factor c2 (i.e., alllengths are stretched by the factor c). Then M and Mc are homeomorphicmanifolds (in particular, kMk = kMck) and

vol(Mc) = cn � vol(M)

where n is the dimension of M. Moreover, Mc is also flat. Therefore bothuniversal covers eM and fMc are isometric to Rn [Lee1, Theorem 11.12]. Thusthe proportionality principle yields

kMk

vol(M)=

kMck

vol(Mc)=

kMk

cn � vol(M).

Since this holds for all c 2 R>0 , clearly kMk = 0 follows.

Since the simplicial volume is a homotopy invariant, we can deduce fromthe proportionality principle that the Riemannian volume in some specialcases is homotopy invariant:

Corollary (5.27). If M and N are homotopy equivalent oriented closed connectedRiemannian manifolds with isometrically isomorphic universal Riemannian cover-ings and kMk 6= 0, then

vol(M) = vol(N).

121


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126

Table of Notation

Symbols# connected sum� one point space_ wedgeh� ,�i Kronecker product 29,

30, 75, 83R

� ! integration of theform ! over themeasure chain � 97

R

�! integration of the form! over the smoothsingular simplex � 95

�k 56

k�k1 `1-norm 5k�k1 seminorm 5k�k

R1 “seminorm” 21

k�ksm1 seminorm on smooth

singular homology 72k�kmh seminorm for measure

chains 62k�k total variation of a

signed measure 51

kMk simplicial volume 5kMkC 21kMkQ 21kMkZ 21k�k

1seminorm 26, 27, 74, 82

k�k1

supremum norm 26, 74[x] (co)homology class

represented by x

[M] fundamental class 5[M]R fundamental class with

real coefficients 5[M]R fundamental class with

R-coefficients 21[M]mh measure homology

fundamental class 80[M] sm fundamental class in

smooth singularhomology 73

AA 85

127

Table of Notation

BB 83, 85B(G,V ) set of bounded

functions from G to V36

BMr (x) open ball in M of

radius r around x 90B

eMr (x) open ball in eM of

radius r around x 90

CC complex numbersC�(X) = C�(X,Z) 3C�(X,R) singular chains of X

with coefficients in R 2C�(f ,R) homomorphism

induced by f 4C�(M) measure chains 61C�(M,N) relative measure chain

group 62C�(X) = C�(X,Z) 3C�(X,R) singular cochains of X

with coefficients in R 3C�(f ,R) homomorphism

induced by f 4bC�(X) set of bounded

cochains in X 26colim colimit/direct limitbCk(M) bounded Borel

measurable functionsmap

1(�k,M) ! R 82

bC�sm (M) bounded smooth

cochains of M 74Csm� (M,R) set of smooth singular

simplices in M 72cX comparison map 30

DD 85@ boundary operator 3,

61, 72deg degree of a map 8�k standard k-simplex 2

Æ coboundary operator3, 74, 82

@j j-th face of a singularsimplex 2

@j j-th face of a standardsimplex 2

@j homorphism inducedby the j-th face map 61

dM metric on M 89d

eMmetric on eM 89

EE 85

FFk (X) set of bounded

functionsin map( eXk+1,R) 44

Fg oriented closedconnected surface ofgenus g 10

F� 114

HH�(X) = H�(X,Z) 4H�(f) = H�(f ,Z) 4H�(X,R) singular homology

of X with coefficientsin R 3

H�(f ,R) homomorphisminduced by f 4

H�(M) measure homologyof M 62

H�(f) homomorphisminduced by f 63

H�(M,N) relative measurehomology group 62

Hsm� (M,R) smooth singular

homology of M 72H�(X) = H�(X)Z 4H�(f) = H�(f ,Z) 4H�(X,R) singular cohomology

of X with coefficientsin R 3

128

Table of Notation

H�(f ,R) homomorphisminduced by f 4

bH� (X) bounded cohomologyof X 26, 38

bH� (f) homomorphisminduced by f 27, 38

bH� (M) bounded measurecohomology of M 82

bH�sm (M) bounded smooth

cohomology 74hM 112

II�(X) set of �1(X)-invariant

functions in F�(X) 44

iM inclusionCsm� (M,R) ! C�(M)

76im imageIsom1 (U) group of isometries,

endowed with theC1-topology 110

Isomco(U) isometry group withthe compact opentopology 102

Isom+1 (U) group of orientation

preserving isometries,endowed with theC1-topology 102

JjM inclusion

Csm� (M,R) ,!

C�(M,R)72

Kker kernel

L�k 56

Mmap (X,Y) set of continuous maps

from X to Y 54mapco

1(M,N) set of all smooth maps

from M to N with thecompact opentopology 54

mapco1(�k,M) 57

map1(M,N) set of all smooth maps

from M to N with theC1-topology 54

map1(M,�k) 56

map1(�k,M) 56

minvol(M) minimal volume 13�f pushforward of � with

respect to f 53

NN nonnegative integers

P�1 fundamental grouppM 100pN 100

QQ rational numbers

RR real numbers

SSk (X) set of singular

k-simplices in X 2s smoothing operator 73secg(M) sectional curvature

of M with respect tothe Riemannianmetric g 13

smearM first step of smearing115

smearN ,M smearing 117

129

Table of Notation

TT�k = �k�Rk 57Tf differential of f

Uu� 45UK 54

VV" 91eV" 91VG G-fixed points in V 37vn maximal volume of an

ideal n-simplex inhyperbolic space 12

vol(M) volume of MvolM volume form of M

ZZ integers

130

Index

Aamenable 39, 39, 43aspherical 17

BBorel function 53Borel �-algebra 52boundary 3boundary operator 3, 61bounded cochain 26bounded cohomology 26

characteristic classes and 42dimension axiom 28duality principle of 29excision 42first group 33fundamental group and 40homotopy invariance of 28Kronecker product for 30naturality of 27of groups 35, 38simplicial volume and 32

smooth 74bounded G-module 35bounded measure cohomology 82bounded smooth cochain 74bounded smooth cohomology 74

duality principle of 76Kronecker product for 75

CC1-topology 54, 58canonical seminorm 38chain

measure 61singular 2smeared 115, 117

characteristic classes 42coboundary 3coboundary operator 3, 74, 82cochain

bounded 26bounded smooth 74singular 3

131

Index

cocycle 3cohomology

boundedsee bounded cohomology

bounded measure 82singular 3

compact open topology 54, 55comparison map 30C1-topology 55connecting homomorphism for mea-

sure homology 64contracting homotopy 36cycle 3

Ddecomposition

Hahn 51Jordan 51

degree 8determination set 50differential 57dimension axiom

bounded cohomology 28measure homology 63

duality 29duality principle

of bounded cohomology 29of bounded smooth cohomology

76of measure homology 84

Eexcision

bounded cohomology and 42for measure homology 64

extension 56

Fface 2flat manifold 121functorial seminorm 24functoriality 8, 23fundamental class

in measure homology 80in singular homology 5in smooth singular homology 73

fundamental theorem of homologicalalgebra 37

GG-module

bounded 35relatively injective 36

G-morphism 35strongly injective 35

G-resolution 36, 37, 38, 44, 45relatively injective 36standard 36strong 36

group actionon map( eXk+1,R) 44on B(G,V ) 36on B(Gn ,R) 36

HHaar measure 112Hahn decomposition 51homology

measure 62singular 3smooth singular 72

homotopy invarianceof bounded cohomology 28of measure homology 64of simplicial volume 9

Hopfian 19

Iinduced measure 53integrable 52integral 52integration

and volume 6measure homology and 98of measure chains 95, 97of smooth singular simplices 6,

95, 95

132

Index

isometric isomorphism 31, 39, 40, 42,45, 72, 75, 80, 81, 85

isometry 102isometry group 102, 102, 103, 105,

110, 110measure on 112

JJordan decomposition 51

KKronecker product 29

for bounded cohomology 30for bounded smooth cohomology

75for measure homology 83, 83

LL2-invariants 17left invariant 112long exact pair sequence for measure

homology 64

Mmapping theorem 42Mayer-Vietoris sequence for measure

homology 65measure chain 61

integration of 95, 97relative 62

measure fundamental domain 112measure homology 60, 62

boundary operator of 61colimits and 66connecting homomorphism 64dimension axiom 63duality principle of 84excision for 64fundamental class 80homotopy invariance of 64integration and 98Kronecker product 83, 83long exact pair sequence 64

Mayer-Vietoris sequence 65naturality of 63relative 62simplicial volume and 81singular homology and 81

minimal volume 13Mostow rigidity theorem vmultiplicativity 8, 9, 19�-negative 50�-positive 50

Nnaturality

of bounded cohomology 27of measure homology 63

negative variation 51normal 65null set 50

Ppositive variation 51proportionality principle 120

applications 121of L2-invariants 19of simplicial volume 13, 100toy version of 12

pushforward of a signed measure 53

Rregular 111relative measure chain group 62relative measure homology 62relatively injective

G-module 36G-resolution 36

Sseminorm 5, 21, 23, 26, 27

bounded measure cohomology82

canonical 38measure homology 62smooth bounded cohomology 74

133

Index

smooth singular homology 72signed measure 50simplex

singular 2smooth singular 72standard 2

simplicial volume 5amenability and 43bounded cohomology and 32classifying maps and 43fudamental group and 43generalisations 21homotopy invariance of 9L2-invariants and 17measure homology and 81of Sn 9of connected sums 44of flat manifolds 14, 121of hyperbolic manifolds 12of products 34of S1 7of surfaces 10, 13proportionality principle 13, 100realisability of 15S1-actions and 15

singular chain 2singular cochain 3singular cohomology 3singular homology 3

measure homology and 81singular simplex 2

integration of 6, 95, 95small 89smearing 101, 114, 115, 117, 117, 118smooth 56smooth singular homology 72smooth singular simplex 72

integration of 6, 95, 95standard resolution 36standard simplex 2strong 36strongly injective 35supremum norm 26, 74

Ttopology

C1-topology 54, 55compact open 54, 55on Isom1 (U) 110on Isom+

1 (U) 102on Isomco(U) 102on map (X,Y) 54on mapco

1(M,N) 54

on map1(M,N) 54

on mapco1(�k,M) 57

on map1(�k,M) 58

rocks!see topology

total variation 51, 51, 52

Uunimodular 112universal Riemannian covering 100

Vvariation 51

negative 51positive 51total 51, 51, 52

volumeminimal 13simplicial

see simplicial volume

134

Documents

The Proportionality Principle of Simplicial Volumehopf.math.purdue.edu/Strohm/diploma_main.pdfample, Gromov’s proof of the Mostow rigidity theorem is based on the fol-lowing proportionality