The Proportionality Principle of Simplicial Volume Clara Strohm Westf"alische Wilhelms-Universit"at Mu"nster Fachbereich Mathematik und Informatik Diplomarbeit The Proportionality Principle of Simplicial Volume Clara Strohm September 2004 Betreuer : Prof. Dr. Wolfgang Lu"ck Zweitgutachter: PD Dr. Michael Joachim Hiermit versichere ich, dass ich diese Arbeit selbst"andig verfasst und nur die angegebenen Quellen und Hilfsmittel benutzt habe. Mu"nster, 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 are concerned with the nature of this interaction between algebraic topology, differential topology and differential geometry. For example all incarnations of the index theorem, the Gauss-Bonnet formula, and the Mostow rigidity theorem fall into this category. Theorem (Mostow Rigidity Theorem). Each homotopy equivalence between oriented closed connected hyperbolic manifolds of dimension at least 3 is homot* *opic to an isometry. This theorem was first proved by Mostow [Mostow ]. But there is also a proof 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 connected manifolds defined in terms of the singular chain complex with real coeffi- cients. This invariant consists of a nonnegative real number measuring the efficiency of representing the fundamental class using singular simplices: If M 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 class can be viewed as a generalised triangulation of the manifold, the simplicial volume can also be interpreted as a measure for the complexity of a mani- fold. But the simplicial volume also has another important interpretation: it is a 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 cl* *osed connected manifold M of dimension n, __kMk___ 1 = ___. vol(M) vn In particular, the simplicial volume does not always vanish. However, in general the simplicial volume is rather hard to compute. In the most cases where the simplicial volume is known, it is zero. This, for example, is the case 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 on hyperbolic 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 other volumes, such as the Riemannian or the minimal volume [Gromov2 ]. Fur- thermore, Gromov conjectured the following relation between the simplicial volume and L2-invariants [Gromov3 , Section 8A] - thereby connecting two distant fields of algebraic topology: Conjecture. Let M be an aspherical closed orientable manifold whose simplicial volume vanishes. Then for all p 2 N, b(2)peM; N (ss1(M)) = 0. A rather complete account of L2-invariants is given in the book L2-Invariants: Theory and Applications to Geometry and K-theory [Lu"ck2]. 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 in Section 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>0 and a constant T 2 R>0 satisfying: for any discrete group G acting freely and cocompactly on U by isometries and all p 2 N, b(2)pU; N (G) = Bp vol(G n U). If U with this G-action is of determinant class, then %(2) U; N (G) = T vol(G n U). In particular, if M and N are oriented closed connected Riemannian man- ifolds with isometrically isomorphic universal Riemannian coverings, then for all p 2 N, (2) b(2)peM;_N_(ss1(M))_ bp eN; N (ss1(N)) = ___________________ vol(M) vol(N) and %(2)_eM;_N_(ss1(M))_ %(2) eN; N (ss1(N)) = ___________________. vol(M) vol(N) Theorem (Proportionality Principle of Simplicial Volume). Let M and N be oriented closed connected Riemannian manifolds with isometrically isomorphic universal Riemannian coverings. Then _kMk____ kNk = _______. vol(M) vol(N) These proportionality principles are also examples for an interesting link between topological and differential structures. A similar proportionality principle is known for Chern numbers [Hirzebruch , Satz 3]. A proof in the L2-case is given in Lu"ck's book [Lu"ck2, Theorem 3.183]. For the simplicial volume, sketches of (different) proofs were given by Gromov [Gromov2 , Section 2.3] and Thurston [Thurston , page 6.9]. However, there is 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 peculiar variant of singular cohomology. Its application as a basic tool for the analysis of the simplicial volume was discovered by Gromov [Gromov2 ]. However, we will mostly refer to the more elegant approach due to Ivanov, based on classical homological algebra [Ivanov1 ], [Monod ]. Thurston's proof of the proportionality principle relies on the computation of the simplicial volume by means of a new homology theory called mea- sure homology [Thurston , pages 6.6-6.9]. Instead of linear combinations of singular simplices, measures on the set of (smooth) singular simplices are considered as chains, thereby permitting averaging constructions such as Thurston's smearing. But Thurston's exposition lacks a proof of the fact that measure homology and singular homology give rise to the same simplicial volume. 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 will be shown that measure homology indeed can be used to compute the simpli- cial volume, i.e., that singular homology with real coefficients is isometrical* *ly isomorphic to measure homology. The proof is based on techniques from bounded cohomology, as explained in Chapter 2. It will also be discussed why 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 isometry groups of Riemannian manifolds. En passant, it will be proved that the compact open topology and the C1-topology (the smooth counterpart of the compact open topology) on the isometry groups coincide. Unfortunately, I was not able to show this without making use of the theory of standard Borel spaces. The chapter closes with a proof of the fact that the simplicial volume of a flat orientable closed connected manifold vanishes. The proof is based on 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. A variety of textbooks is available providing such a background: e.g., [Lee1 ], [Lee2 ], [Massey ], [Bredon ], [Elstrodt]. But the detailed exposition of the s* *ub- 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 nonnegative integers, the set of rational numbers, the set of real numbers, and the set of complex numbers respectively. The one point space is written as ffl. For simplicity, 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 * *in the table of notation (between bibliography and index). Acknowledgements _________________________________________________________________________ First of all I would like to thank Professor Wolfgang Lu"ck for giving me the opportunity to write this diploma thesis, for the encouragement, and for the freedom I enjoyed during this project. Roman Sauer and Marco Schmidt infected me with their fascination for the simplicial volume and bounded cohomology. I am very grateful for all their support and guidance. Marco Varisco was always open for discussions on mathematical 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 led me through my two years at the University of Konstanz. I would like to thank him for all his support. Furthermore, I would like to thank Stefanie Helker for studying English and Benedikt Plitt, Julia Weber (Wohlfu"hlbu"ro!), and Michael Weiermann for hunting bugs and for their valuable suggestions. My parents always supported me, even though I did not manage to become acquainted with philosophy - but at least I am no #gewm#trhtoc anymore. Dankesch"on! All my gratitude is devoted to Andres L"oh, not only for his TEXnical eksper- tise, but also for his patience and all the things language is not powerful enough to express - except for Plu"schtiere: Wuiisch? Zonder pf! ix Contents _________________________________________________________________________ 1___Simplicial_Volume______________________________________________1_____ 1.1 Singular (co)homology 1.3 A zoo of properties 7 refreshed 2 1.4 Simplicial volume and 1.2 Definition of the L2-invariants 17 simplicial volume 4 1.5 Generalisations 21 2___Bounded_Cohomology________________________________________25_________ 2.1 Bounded cohomology 2.3 Bounded cohomology of spaces 26 of groups 35 2.4 The mapping theorem 40 2.2 Duality 29 2.5 A special resolution 44 x 3___Measure_Homology___________________________________________49________ 3.1 Prelude 50 3.3 Basic properties of 3.2 Definition of measure measure homology 63 homology 60 4___Measure_Homology_is_Singular_Homology_________________71_____________ 4.1 Smooth singular 4.3 . . . is isometric 80 homology 72 4.2 The algebraic 4.4 Integrating measure isomorphism 76 homology chains 95 5___The_Proportionality_Principle________________________________99______ 5.1 Statement of the 5.5 Proof of the proportionality proportionality principle 100 principle 120 5.2 Isometry groups 102 5.6 Applications of the 5.3 The Haar measure 111 proportionality 5.4 Smearing 114 principle 121 xi xii 1 Simplicial Volume _________________________________________________________________________ The simplicial volume of an oriented closed connected manifold is a homotopy invariant - defined in terms of the singular chain complex - which measures the efficiency of representing the fundamental class by singular chains (with real * *coef- ficients). Its first appearance is in Gromov's famous proof of the Mostow rigid* *ity theorem [Munkholm ]. Despite of being a topological invariant the simplicial volume contains interes* *ting information about the possible differential structures on a manifold - in form * *of the Riemannian volume. Examples for such relations are Gromov's estimate of the minimal volume and the proportionality principle, which is the topic of this th* *esis. 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 * *volume vanishes. Then for all p 2 N, b(2)peM; N (ss1(M)) = 0. In this chapter the simplicial volume is introduced (Section 1.2) and some of i* *ts properties are collected (Section 1.3). We then give a short survey about indic* *ations supporting the above conjecture in Section 1.4. In the last section we briefly * *discuss some 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 will be used throughout this thesis. The key objects of singular (co)homology are simplices: Definition (1.1). For k 2 N the set ae fifik oe k := (x0, : :,:xk) 2 Rk+1~0fifiXxj= 1 j=0 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 the notation since it can easily be inferred from the context. The main idea of the singular theory is to examine all continuous maps from the standard simplices to a topological space to detect the "holes" of this space, i.e., to measure which "cycles" are no "boundaries". Definition (1.2). Let X be a topological space and k 2 N. o The set Sk(X) of singular k-simplices is the set of all continuous func- tions k ! X. If oe 2 Sk(X) and j 2 f0, : :,:kg, then oe ffi @jis called* * the j-th face of oe. o 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 generated by Sk(X) . 2 1.1 Singular (co)homology refreshed o The boundary operator @: Ck+1 (X, R) ! Ck(X, R) is the R-module homomorphism given on the generators oe 2 Sk+1 (X) by k+1 @(oe) := X ( 1)j oe ffi @j. j=0 The elements of im @ are called - as geometry suggests - boundaries, and the elements of ker@ are called cycles. o 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):= Hom Z Ck(X, Z), R = Hom R Ck(X, R), R o The boundary operator @: Ck+1 (X, R) ! Ck(X, R) induces the co- boundary operator ffi :Ck (X, R) ! Ck+1 (X, R)as follows: ffi :Ck (X, R)! Ck+1 (X, R) k+1 f 7 ! c 7! ( 1) f(@c) . The elements of im ffi are called coboundaries and the elements of kerffi are called cocycles. In the special case R = Z, we use the abbreviations Ck(X) := Ck(X, Z) and Ck (X) := Ck (X, Z). A lengthy calculation shows that in the situation of the above definition the equality @ ffi @ = 0 holds (i.e., the singular chains form a chain complex). Therefore ffi ffi ffi = 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. o The R-module ker @: Ck(X, R) ! Ck 1 (X, R) Hk(X, R) := _______________________________ im @: Ck+1 (X, R)! Ck(X, R) is called k-th singular homology of X with coefficients in R. o The R-module ker ffi :Ck (X, R)! Ck+1 (X, R) Hk (X, R):= _______________________________ im ffi :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 ae M, and (ab)b2B ae R. When we write P b2Bab b, we implicitly assume that all but a finite number of the coefficients (ab)b2B are 0 unless stated otherwise. Additionally, whenever we write something like "Let P oe2Sk(X)aoe oe 2 Ck(X, R)." it is understood that all aoeare 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 aoe oe7 ! X aoe f ffi oe oe2Sk(X) oe2Sk(X) and Ck (f, R):Ck (Y, R)7 ! Ck (X, R) g7 ! g ffi Ck(f, R). The corresponding maps on (co)homology are denoted by H (f, R)and H (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 Definition 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 singular chains a = P oe2Sk(X)aoe oe 2 Ck(X, R) we define kak1 := X jaoej. oe2Sk(X) Since the sum is finite, this norm is well-defined and turns the singular chain group Ck(X, R) into a normed real vector space. Definition (1.7). Let X be a topological space, k 2 N and ff 2 Hk(X, R). Then k k1induces a seminorm as follows: fi kffk1:= inf kak1 fia 2 Ck(X, R), @(a) = 0, [a] = ff . In general, this is not a norm since the set of boundaries does not need to be 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) ,= Z which is compatible with the 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 M by [M] 2 Hn (M) . Moreover, we write [M]R for the image of [M] under the change of coefficients homomorphism Hn (M, Z) ! Hn (M, R). Definition (1.9). The simplicial volume of an oriented closed connected manifold M is defined as fl fl kMk := fl[M]Rfl1. Since k k1is not affected by multiplication with 1, the simplicial volume is 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 oe1, : :,:oem : n ! M is a tri- angulation of M, then P mj=1oej represents [M]. Hence the simplicial volume measures 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 definition to 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 Poe2Sn(M)aoe oe 2 Cn (M, R) is a cycle representing [M]R consist* *ing of smooth singular simplices, then Z vol (M) = X aoe volM . oe2Sn(M) oe 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 oe1, : :,:oem : n ! M [Whitehead ]. Then one shows that the lemma is true for the cycle P mj=1oej (which is intuitively clear) and uses the fact that the integral only depends_on the represented homology class [Lee2 , Exercise 16.2 and Theorem 16.10]. |__| Stronger connections - such as the proportionality principle - between the simplicial volume and other volumes are discussed in Subsection 1.3.2 and Chapter 5. As a first example we will give a direct calculation for kS1k. There are much more sophisticated techniques for computing kS1k, but the following proof emphasises geometric intuition. Theorem (1.11). Let X be an arcwise connected topological space. Then kffk1= 0 holds for all ff 2 H1 (X, R). Proof.The universal coefficient theorem yields H1 (X) R ,=H1 (X, R). In other words: if ff 2 H1 (X, R), then there exist k 2 N, r1, : :,:rk 2 R, and ff1, : :,:ffk 2 H1 (X) such that ff corresponds to k X rj ffj. j=1 6 1.3 A zoo of properties By the Hurewicz theorem each of the ffjcan be represented by a continuous function oej: S1 ! X. For each j 2 f1, : :,:kg and each d 2 N~1 we write oe(d)jfor the composition ("wrapping oejaround the circle d times") oejffi (z 7! zd): S1 ! X. Now the Hurewicz theorem tells us that [oe(d)j] = d [oej] holds in H1 (X) (and thus also in H1 (X, R)). In particular, fl (d)fl flk oej fl 1 k kffk1~ flflXrj ____flfl~ __ X jrjj. j=1 d 1 d j=1 __ Since this is true for any d 2 N~1, we get kffk1= 0. |__| __ Corollary (1.12). In particular, kS1k = 0. |__| This corollary also implies that the seminorm k k1on H1 S1, R is not a norm. It is more difficult to show that there are manifolds with non-zero simplicial volume. For example, the simplicial volume of hyperbolic manifolds does not 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 the definition. We then give a collection of more sophisticated properties, based on the comparison with other volumes and on the theory of bounded coho- mology. This section ends with a short (and rather incomplete) discussion of 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 compute the simplicial volume in some special cases. 7 1 Simplicial Volume We start with a simple - but fundamental - observation. Gromov calls the described property functoriality [Gromov4 , 5.34]. Lemma (1.13). Let f :X ! Y be a continuous map of topological spaces, k 2 N and ff 2 Hk(X, R). Then we get for the induced seminorm kHk(f, R)(ff)k1~ kffk1. Proof.Let Poe2Sk(X)aoe oe 2 Ck(X, R) be a cycle representing ff. Then the class Hk(f, R)(ff) is represented by P oe2Sk(X)aoe f ffi oe. Therefore fl~ ~ fl fl fl fl fl fl fl kHk(f, R)(ff)k1 = flflX aoe f ffi oe flfl~ flflX aoe f ffi oeflfl oe2Sk(X) 1 oe2Sk(X) 1 fl fl fl fl ~ X jaoej = flflX aoe oeflfl. oe2Sk(X) oe2Sk(X) 1 Taking the infimum over all cycles representing ff yields the desired inequal-_ ity kHk(f, R)(ff)k1~ kffk1. |__| 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-dimensional manifolds 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 as the 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 manifolds of the same dimension. Then kMk ~ j deg(f)j kNk . Proof.Using the previous lemma and the above remark we conclude that fl fl kMk = fl[M]Rfl1 fl fl ~ flH (f, R)([M] R)fl1 fl fl = fldeg(f) [N]Rfl1 __ = j deg(f)j kNk . |__| 8 1.3 A zoo of properties Corollary (1.17). Since homotopy equivalences between oriented closed connected manifolds are maps of degree 1 or 1, the simplicial volume is a homotopy_invar* *iant. |__| Corollary (1.18). If f :M ! M is a self-map of an oriented closed connected manifold with j deg(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 follows that 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 oe 2 Sn (N), let L(oe) be the set of singular simplices eoe2 Sn (M) with p ffi eoe= oe. Since p is a d-sheeted covering and n is arcwise con- nected and simply connected, the set L(oe) contains precisely d elements. If a = P oe2Sn(N)aoe oe 2 Cn (N, R)represents [N]R , then ea:= X aoe X eoe oe2Sn(N) eoe2L(oe) represents [M]R because @(ea) = 0, ~ ~ Hn (p, R)[ea] = X aoe d oe = d [N]R oe2Sn(N) = Hn (p, R)[M] R , and deg(p) = d 6= 0. Thus fl fl kMk = fl[M]Rfl1~ keak1~ d X jaoej = d kak1, oe2Sn(N) fl fl __ which implies kMk ~ d fl[N]Rfl1= d kNk . |__| 9 1 Simplicial Volume Using this result we can apply covering theory to obtain an upper bound for 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 of genus g. Then fl fl flFgfl~ 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 be triangulated by 4g 2 simplices: This triangulation induces a "triangulation" of Fg consisting of 4g 2 sim- plices oe1, : :,:oe4g 2. This is no triangulation of Fg in the strict sense, bu* *t its second barycentric subdivision is. Hence we get [Bredon , page 338f] that 4g 2 P j=1 oejis a cycle representing Fg R, which yields the inequality fl fl __ flFgfl~ 4g 2. |__| One can now combine the above inequality with a beautiful covering theory argument 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 1.3 A zoo of properties with connected covering space: we have [Massey , Example IV 5.3] ss1(Fg)ab,=Z2g, where ss1(Fg)ab is the abelianisation of ss1(Fg). Thus there is a surjective group homomorphism ss1(M) ,=ss1(Fg) ! ss1(Fg)ab ,=Z2g ! Z. The inverse image under this homomorphism of a subgroup of Z of index d is a subgroup of ss1(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 by using a CW-decomposition (or by calculating the volume if the manifolds are hyperbolic), O(Md) = d O(M). Thus we get for the genus O(Md) + 2 d O(M) + 2 g(Md) = ____________ = ______________ 2 2 = 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 1 1 fl fl kMk = __ kMdk = __ flFg(M )fl d d d 1 ~ __ 4g(Md) 2 d 2 = 4g 4 + __. d This inequality holds for all d 2 N~2. Therefore it follows that __ k Mk ~ 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 finitely many sheets. Then _k_Mk___ kNk = _______. vol(M) vol(N) Proof.In this situation, both the (Riemannian) volume and the simplicial volume 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 given in Chapter 5, based on the results about measure homology in Chapters 3 and 4. 1.3.2 Other volumes There is a great variety of connections between the simplicial volume and other volumes arising in differential geometry. We have already seen a dis- tant relation between the simplicial volume and the Riemannian volume in Lemma (1.10). Gromov used the following fact about hyperbolic manifolds in 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 cl* *osed connected manifold M of dimension n, __kMk___ 1 = ___. vol(M) vn Proof.The estimate kMk__volM~ _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 1.3 A zoo of properties Example (1.24). Let M be an oriented closed connected surface of genus g 2 N~2. In view of the homotopy invariance of the simplicial volume we may assume 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) = 2ss O(M) = 2ss (2g 2). Moreover, v2 = ss [Benedetti, Petronio, Lemma C.2.3]. Thus Theorem (1.23) implies k Mk = 4g 4. In particular, the simplicial volume can grow arbitrarily large and it is no bordism invariant. Based on straightening, Inoue and Yano found the following generalisation of Theorem (1.23) [Inoue, Yano ]: Theorem (1.25). For each n 2 N there is a constant Cn 2 R>0 such that the following property holds: Let ffi 2 R>0 and let M be an n-dimensional oriented closed connected Riemannian manifold whose sectional curvature is bounded from above by ffi. Then kMk ~ Cn ffin/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____ kNk = _______. vol(M) 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 satisfies j 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~2 is 2ss 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 by Berger [Gromov2 , page 215]. According to Gromov, the minimal volume can be estimated from below as follows [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_mani* *fold, then kMk = 0. |__| We will prove this corollary as an application of the proportionality principle in 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 can be 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 kerss1(f) is amenable, then k Mk = kNk . The class of amenable groups contains, for example, all Abelian groups and all finite groups, but it does not contain the free product Z Z. In particula* *r, we see again that kSnk = 0 for all n 2 N>0 and that kS1 S1k = 0. More details will be given in Sections 2.3.3 and 2.4.3. 14 1.3 A zoo of properties Other consequences of the duality between k k1and bounded cohomology are (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 that the following holds: if M and N are two oriented closed connected manifold* *s, 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 k M #Nk = kMk + kNk . In particular, for all oriented closed connected manifolds M, the simplicial volume of the product S1 M is zero. This is generalised in the following way [Yano ]: Theorem (1.33). Suppose M is an oriented closed connected manifold admitting a non-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 simplicial volume 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-complexes consisting of finitely many cells. Proof.We prove by induction (on k 2 N) that there are only countably many homotopy 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 let Y 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 and the 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 ssn(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-complexes __ with k + 1 cells. |__| Corollary (1.35). The set fi kMk fiM is an oriented closed connected manifoldae 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]. By the previous theorem, there are only countably many homotopy types of closed (oriented, connected) manifolds. Now Corollary (1.17) completes the_ proof. |__| 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. It would be interesting to know whether the simplicial volumes lie densely in R>0. It would be sufficient to know that there is a sequence (Mn)n2N of oriented closed connected manifolds (of dimension at least 3) with non-zero simplicial volume such that lim kMnk = 0. n!1 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 subset of R. This speculation is based on a result of Thurston on hyperbolic mani- folds of dimension 3 [Gromov1 ], [Thurston ]. However, until today there do not 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 numbers and L2-torsion. More concretely, Gromov formulated the following conjec- ture [Gromov3 , Section 8A]: Conjecture. Let M be an aspherical closed orientable manifold whose simplicial volume vanishes. Then for all p 2 N, b(2)peM; N (ss1(M)) = 0. Lu"ck extended this conjecture as follows [Lu"ck1, Conjecture 3.2]: Moreover, eM is of determinant class and %(2) eM; N (ss1(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 is contractible. (For CW-complexes this is the same as to require that all higher homotopy 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 [Lu"ck2, Chap- ter 6]. Until now there does not seem to exist a strategy to prove (or disprove) the above conjecture. However, there are many facts witnessing the probable truth of this conjecture. In the following paragraphs we will present some of this evidence. A more extensive list can be found in Lu"ck's book [Lu"ck2, Chapter 14]. For simplicity we refer to the proofs in this book instead to the original 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 zeroth L2-Betti number: Lemma (1.37). Let M be an aspherical connected closed manifold. Then the funda- mental group ss1(M) is torsion free and hence infinite. Proof.Assume that ss1(M) contains a finite cyclic subgroup G. Since M is aspherical and of finite dimension n, the cellular chain complex Ccell(Me; Z) is a bounded projective resolution of the trivial ZG-module Z. This implies that the group homology cell H (G; Z) ,=H C (Me; 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., ss1(M) is torsion free. |_* *_| Corollary (1.38). If M is an aspherical closed connected manifold, then [L"uck2, Theorem 6.54(8)] 1 __ b(2)0eM; N (ss1(M)) = ________ = 0. |__| jss1(M)j Example (1.39). The condition that the manifold be aspherical in Gromov's conjecture is necessary: The manifold S2 is closed and orientable, but not aspherical because ss2(S2) ,= Z. By Corollary (1.18) we have kS2k = 0. On the other hand [Lu"ck2, Theorem 6.54(8)], 2 1 b(2)0eS2; N (ss1(S )) = ________ = 1 6= 0. jss1(S2)j 1.4.2 Amenability If M is on oriented closed connected (not necessarily aspherical) manifold with amenable fundamental group, then kMk = 0 (Corollary (2.32)). The corresponding result in the L2-world is: Theorem (1.40). Let M be an oriented closed connected aspherical manifold with amenable 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. But then it can be shown that all L2-Betti numbers of M must be zero [Lu"ck2,__ Corollary 6.75]. |__| 1.4.3 Proportionality One of the most remarkable similarities between the simplicial volume and L2-invariants is the proportionality principle. It is valid for the simplicial volume (Theorem (5.1)) and for L2-Betti numbers (and L2-torsion) [Lu"ck2, 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 2 R>0 and a constant T 2 R>0 satisfying: for any discrete group G acting freely a* *nd cocompactly on U by isometries and all p 2 N, b(2)pU; N (G) = Bp vol(G n U). If U with this G-action is of determinant class, then %(2) U; N (G) = T vol(G n U). 1.4.4 Multiplicativity As we have seen in Section 1.3, the simplicial volume is multiplicative with respect to the mapping degree. Therefore Gromov's conjecture predicts that if 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. This statement is known to be true if this self-map is a covering [Lu"ck2, 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 epimorphism G ! 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 be seen by considering the shift operator. 19 1 Simplicial Volume Theorem (1.43). Let M be an aspherical oriented closed connected manifold. If there is a continuous map f :M ! M such that j deg(f)j ~ 2 and if all sub- groups of ss1(M) of finite index are Hopfian, then 8p2N b(2)peM; N (ss1(M)) = 0. A proof is given in [Lu"ck2, Theorem 14.40]. (There only all normal subgroups of ss1(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 [Lu"ck2, Corollary 1.43]: Theorem (1.44). Let M be an aspherical closed smooth manifold with non-trivial S1-action. Then 8p2N b(2)peM; N (ss1(M)) = 0. The corresponding statement about the simplicial volume is Theorem (1.33). 1.4.6 Gromov's conjecture Looking at the conjecture and the known evidence one could also suspect that the converse is true: If M is an aspherical oriented closed connected manifold whose L2-Betti numbers vanish, then kMk = 0. But this statement is false: If M is an oriented closed connected hyperbolic manifold of odd dimension, then b(2)peM; N (ss1(M)) = 0 for all p 2 N [Lu"ck2, 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 are described. 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 Poincar'e complexes. In the last section we consider functorial seminorms in the sense of Gromov. 1.5.1 Unreal coefficients Instead of real coefficients we could also use the singular chain complex with integral, complex or rational coefficients to define "simplicial volumes" kMk Z, kMk C , and kMk Q . 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 the image 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 aoe oe 7 ! X jaoej, oe2Sn(M) oe2Sn(M) which induces a "seminorm" k kR1on Hn (M, R) by taking the infimum over all representatives. That is, we define R fi kffkR1:= inf kak1 fia 2 Cn (M, R), @a = 0, [a] = ff 2 Hn (M, R) for all ff 2 Hn (M, R). In particular, we write fl flZ fl flC fl flQ kMk Z := fl[M]Zfl1, kMk C := fl[M]Cfl1, kMk Q := fl[M]Qfl1. 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 integral simplicial volume has an interesting connection with the Betti numbers of the 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 manifold and P tj=1aj oej2 Cn (M, Z) represents [M]Z . Then n+1 X bk(M) ~ t 2 , k2N where bk(M) = rk Hk(M) is the k-th Betti number of M (which is the same as dim RHk(M, R) and dim Q Hk(M, Q) ). In particular, n+1 X bk(M) ~ 2 kMk Z . k2N The proof is based on Poincar'e duality [Lu"ck2, 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 connected Riemannian manifold with k1 ~ sec(M) ~ k2 for some k1 ~ k2 > 0, then the Betti numbers of M can be estimated via X bk(M) ~ C(n, k1/k2) kMk . k2N It is not hard to see that the real, complex, and rational simplicial volume all coincide: Lemma (1.48). Let M an oriented closed connected manifold of dimension n. Then kMk C = kMk = kMk Q . Proof.The inclusions C (M, Q) ae C (M, R) ae C (M, C) yield kMk C ~ kMk ~ kMk Q . 22 1.5 Generalisations The inequality kMk C ~ kMk can be shown by considering the real part of a representing cycle. The inequality kMk ~ kMk Q can be obtained via an approximation of boundaries with real coefficients by boundaries with ra- tional coefficients. In both cases one has to use the fact that the fundamental class with rational or complex coefficients comes from a class in homology __ with 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 manifold which is not orientable, we can define 1 fl__fl kMk := __ flMfl, 2 ___ 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 to define a (possibly infinite) simplicial volume. If M is a manifold with boundary, then we can define the simplicial volume as 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 Poincar'e complexes, 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 induced by the `1-norm, one could also use other seminorms. To ensure that the resulting invariant has useful properties, Gromov introduced the following notion [Gromov4 , 5.34]: 23 1 Simplicial Volume Definition (1.49). Suppose for each topological space X a seminorm k k is given on H (X, R). Such a seminorm is called functorial if for each contin- uous map f :X ! Y and all ff 2 H (X, R) kH (f, R)(ff)k ~ kffk. By Lemma (1.13) the seminorm k k1on homology with real coefficients in- duced by the `1-norm on the singular chain complex is functorial. Other examples are given in Gromov's book [Gromov4 , pages 302 316]. The proof of Corollary (1.16) of course shows that each functorial seminorm is submultiplicative with respect to the degree of maps. Definition (1.50). If k k0is 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 zero for 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 coefficients vanishes for simply connected spaces. Until today, neither a proof nor counterexamples to this amazing conjecture are known. 24 2 Bounded Cohomology _________________________________________________________________________ Bounded cohomology is the functional analytic twin of singular cohomology - it is constructed similarly to singular cohomology from the singular chain complex, using the topological dual space instead of the algebraic dual space [Gromov2 ]* *. The corresponding duality on the level of (co)homology is explored in Section 2.2. * *Using this duality, it is possible to calculate the simplicial volume via bounded coh* *omol- ogy. It is astonishing that the rather small difference between the definitions of s* *ingular and 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 * *source of the strength of bounded cohomology. In Section 2.4, a key feature of bounded cohomology is discussed: the bounded cohomology of a topological space coincides with the bounded cohomology of its fundamental group. Unfortunately, unlike singular cohomology bounded coho- mology fails to satisfy the excision axiom. Nevertheless, in many cases bounded cohomology can be calculated directly from special resolutions. For example, it* * can be shown that bounded cohomology ignores amenable groups, implying that the bounded cohomology of spaces with amenable fundamental group vanishes. In the last section, we will explain a special resolution for calculating the b* *ounded cohomology of a group, which plays a crucial r^ole in Section 4.3. 25 2 Bounded Cohomology 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 complex instead of the algebraic one. The corresponding norm for singular cochains is therefore the supremum norm: Definition (2.1). Let X be a topological space and k 2 N. For a singular cochain f 2 Ck (X, R)the (possibly infinite) supremum norm is defined via kfk 1 := sup jf(oe)j. oe2Sk(X) This induces a seminorm on Hk (X, R)by fi k 8'2Hk(X,R) k'k 1 := inf kfk1 fif 2 C (X, R), ffi(f) = 0, [f] = ' . We write fi bCk(X) := f 2 Ck (X, R)fikfk1 < 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., k k+1 ffi Cb (X) ae bC (X). Thus bC (X) is a cochain complex. Definition (2.2). Let X be a topological space and k 2 N. The k-th bounded cohomology group of X is defined by ker ffij bk :Cbk(X) ! bCk+1(X) bHk(X) := _________C_(X)____________________. im ffijCbk 1(X):bCk 1(X) ! bCk(X) As for the simplicial volume, it is not immediately clear from the definition that 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 supremum norm on bCk(X) induces a seminorm on bHk(X) by fi k 8'2Hbk(X) k 'k1 := inf kfk1 fif 2 bC (X), ffi(f) = 0, [f] = ' . Except for some special cases it is not known whether this seminorm is a norm [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 least bounded 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 a continuous map. For each k 2 N, the map f induces a homomorphism Hbk(f) :bHk(Y) ! bHk(X) [g]7 ! c 7! g(f ffi c) of operator norm at most 1. Here composition of f with elements of Ck(X, R) is defined by linear extension of composition with elements of Sk(X) . Proof.Let g 2 bCk(Y)with ffi(g) = 0. Since g is bounded on Sk(Y) , the cochain Ck(X, R) ! R c7 ! g(f ffi 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 depend on the chosen representatives g, the map bHk(f)is indeed well-defined. It is clear that bHk(f)is a homomorphism of real vector spaces and that fl fl flbHk(f)[g] fl ~ kgk . 1 1 Taking the infimum over all representatives yields fl fl __ flbHk(f)[g] fl ~ k[g]k . |__| 1 1 27 2 Bounded Cohomology 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 space* *s, then the induced maps are equal: Hb (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., @ ffi k +1 + k ffi @ = C +1 (f, R) C +1 (g, R). This chain homotopy induces the map h : C (Y, R) ! C 1 (X, R) c7 ! ( 1) c ffi k 1. Obviously, h maps bounded cochains to bounded cochains, thus it restricts to bh := h j b :Cb (Y) ! bC 1(X). C (Y) Since +1 +1 +1 ffi ffi bh+ bh ffi=ffif(c)fi ( 1) c ffi k 1 + bh ( 1) c ffi @ = c ffi k 1 ffi @ + c ffi @ ffi k = c ffi C (f, R) C (g, R) = bC (f) bC(g) (c) holds for all c 2 Cb (Y), it follows that bh is a cochain homotopy between bC (f)and bC (g). Hence we get bH (f) = bH (g). |___| Lemma (2.6) (Dimension Axiom). Bounded cohomology satisfies the dimension axiom, i.e., Hb0 (ffl),=R, and 8k2N~1 Hbk(ffl)= 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 k1and k k1 on singu- lar homology and bounded cohomology respectively is explored. Using the Hahn-Banach theorem we will see that the duality on the level of (co)chain complexes 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 bounded cohomology: bounded cohomology can be used to compute the simplicial volume. Since bounded cohomology is much better understood (in view of the 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 and connected sums of manifolds. Moreover, duality will play a central r^ole 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 ff 2 Hk(X, R). 1. Then kffk1= 0 if and only if 8'2Hbk(X) h', ffi= 0. 2. If kffk1> 0, then __1__ fi k = inf k'k 1 fi' 2 bH (X), h', ffi= 1. kffk1 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 with unit 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 ff [f], [a] := f(a). 29 2 Bounded Cohomology In fact, the relations between the coboundary and the boundary operator imply that the Kronecker product is well-defined. Definition (2.9). Let X be a topological space. The homomorphsim cX :Hb (X) ! H (X, R) induced by the inclusion bC (X) ae 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) and Example (2.29)). Definition (2.10). Let X be a topological space, k 2 N, ' 2 Hbk(X) , and ff 2 Hk(X, R). Then the Kronecker product of ' and ff is defined to be ff h', ffi:= cX ('), ff . Proof (of Theorem (2.7)).Let ff be represented by the cycle a = X aoe oe 2 Ck(X, R). oe2Sk(X) Assume that there is a ' 2 Hbk(X) , represented by the bounded cochain f 2 bCk(X), with h', ffi= 1. Then, by definition, fi ` ' fi fi fi 1 = jh', ffij = fifif X aoe oe fifi~ X jaoej jf(oe)j oe2Sk(X) oe2Sk(X) ~ kfk1 kak1. If we take the infimum over all representatives of ' and ff, it follows that 1 ~ k'k 1 kffk1. In particular, if kffk1= 0, there can be no such '. Now, let kffk1> 0, and let a 2 Ck(X, R) be a cycle representing ff. We de- note the closure in Ck(X, R) with respect to the norm k k1by __1. Thus the condition kak1 ~ kffk1> 0 implies _______________________________1 a /2im @: Ck+1 (X, R)! Ck(X, R) . 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, fj______________________1= 0, im @:Ck+1(X,R)!Ck(X,R) fl fl 1 flffl = _____. 1 kak 1 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 ff ff [f], ff = [f], [a] = f(a) = 1 and 1 1 k[f]k1 ~ kfk1 = _____~ _____. kak 1 kffk1 In particular, there exists a ' 2 bHk(X) with h', ffi= 1 and k'k 1 ~ 1/ kffk1. Hence, if kffk1> 0, _1___ fi k __ = inf k'k 1 fi' 2 bH (X), h', ffi= 1. |__| kffk1 Corollary (2.11). Let f :X ! Y be a continuous map such that the induced map Hb (f) :bH (Y) ! bH (X) is an isometric isomorphism. Then fl fl kffk1= flH (f, R)(ff)fl1 for all ff 2 H (X, R). Proof.Since bH (f) is an isometric isomorphism, we obtain fi n ff k'k 1 fi' 2 bH (Y), ', Hn (f, R)(ff) = 1 fl n fl fi n n ff = flbH (f)(')fl1 fi' 2 bH (Y), bH (f)('), ff = 1 fi n = k_k 1 fi_ 2 bH (X) , h_, ffi= 1. __ Therefore application of Theorem (2.7) proves the Corollary. |__| 31 2 Bounded Cohomology 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, then __1__ fl Rfl = fl[M] fl , kMk 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. R ff Proof.By definition, [M] , [M]R = 1. Furthermore, Hn (M, R) ,=R [M]R . Hence im(cnM) 6= 0 if and only if [M]R 2 im(cnM). 1. Therefore the first part of Theorem (2.7) yields that kMk = k[M] Rk1 = 0 if and only if cnM is trivial. 2. Suppose kMk > 0. By the first part, cnM: bHn(M) ! Hn (M, R) ,= R is not trivial and hence surjective. Moreover, fi n n k'k 1 = inf kfk1 fif 2 bC (X), ffi(f) = 0, [f] = ' in bH (M) fi n n n ~ inf kfk1 fif 2 C (X, R), ffi(f) = 0, [f] = cM (') in H (M, R) = kcnM(')k1 holds for all ' 2 Hbn (M) . Using the surjectivity of cnM, we obtain (by the second part of Theorem (2.7)) _1___ 1 = _________flfl kMk fl[M]Rfl1 fi n = inf k'k 1 fi' 2 bH (M) , h', [M]Ri= 1 n fi n n ff ~ inf kcM (')k1 fi' 2 bH (M) , cM ('), [M]R = 1 fi n R ~ inf k_k 1 fi_ 2 H (M, R) ,=[M] R, h_, [M]Ri= 1 fl Rfl = fl[M] fl1. 32 2.2 Duality fl Rfl In particular, fl[M] fl1is finite. On the other hand, for each " 2 R>0 there is a cocycle f 2 Cn (M, R) with [f] = [M]R in Hn (M, R) and fl Rfl kfk1 ~ fl[M] fl1+ " < 1. Thus f 2 bCn(M) . Let ' 2 bHn(M) be the corresponding bounded cohomol- ogy class. Then fl fl k'k 1 ~ kfk1 ~ fl[M]Rfl1+ " and R ff h', [M]Ri= hcnM('), [M]Ri= h[f], [M]Ri= [M] , [M]R = 1, which implies __1__ 1 fl Rfl __ = _________flfl~ k'k 1 ~ fl[M] fl + ". |__| kMk fl[M]Rfl1 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 bounded cohomology 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 arcwise connected. Since the singular homology of a space is the direct sum of the singular homology groups of its path components, this result carries over to all spaces X. Now let ' 2 bH1(X) and let f 2 bC1(X) be a cocycle representing '. Since the seminorm on H1 (X, R)is trivial (Theorem (1.11)), we get from Theorem (2.7) that 8ff2H1(X,R)0 = h', ffi. By the universal coefficient theorem, H1 (X, R) ! hom H1 (X), R _ 7 ! ff 7! h_, ffi 33 2 Bounded Cohomology is an isomorphism. Hence the comparison map c1X:Hb1(X) ! H1 (X, R) must be trivial. Thus there is a cochain u 2 C0 (X, R)with ffi(u) = f. However, in general u is not bounded. Altering u, we construct a bounded cochain bu2 bC0(X) satisfying ffi(bu) = f : Let (X)i2Ibe the path components of X. For each i 2 I we choose a point xi2 Xi. If x 2 X, then there is exactly one i(x) 2 I such that x 2 Xi(x). We define bu2 C0 (X, R)to be the linear extension of 8x2X ub(x) := u(x) u(xi(x)). Since x and xi(x)are in the same path component, there is a path oex in X joining these points. So fi fi fi fi fibu(x)fi= fiu(oex(0)) u(oex(1))fi fi fi = fiu(@(oex))fi fi fi = fiffi(u)(oex)fi fi fi = fif(oex)fi ~ kfk1 , which implies bu2 bC0(X). Furthermore we have for all oe 2 S1 (X) ffi(bu)(oe)= bu(oe(1) oe(0)) = bu(oe(1)) bu(oe(0)) = u(oe(1)) u(xi(oe(1))) u(oe(0)) + u(xi(oe(0))). Here i(oe(1)) = i(oe(0)) because oe(1) and oe(0) lie in the same path compo- nent. Hence ffi(bu)(oe)= u(oe(1)) u(oe(0)) = u( @(oe)) = ffi(u)(oe) = f(oe). __ This proves ' = [f] = 0 in bH (X). |__| 2.2.3 Products of manifolds Bounded cohomology makes it possible to prove that the simplicial volume is 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 the following 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 the cross-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 and Petronio [Benedetti, Petronio, Theorem F.2.5]. The constant c(n) is related to the number of n-simplices needed to triangulate the products p n p for all p 2 f1, : :,:ng. 2.3 Bounded cohomology of groups _________________________________________________________________________ In his paper "Foundations of the Theory of Bounded Cohomology," Ivanov developed a transparent derived functor type approach to bounded coho- mology [Ivanov1 ]. Just like Ivanov, we will consider bounded cohomology of 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 group is based upon special kinds of resolutions. The basic objects are relatively injective bounded G-modules. Definition (2.15). Let G be a discrete group. o A bounded G-module is a Banach space V with a (left) G-action such that kg vk ~ kvk for all g 2 G and all v 2 V . o A G-morphism of bounded G-modules is a bounded G-equivariant linear operator. o An injective G-morphism i :V ! W between bounded G-modules is said to be strongly injective if there is a bounded linear operator 35 2 Bounded Cohomology s: W ! V satisfying s ffi i = idand ksk ~ 1 (but s does not need to be G-equivariant). o A bounded G-module U is relatively injective if for any strongly in- jective G-morphism i :V ! W and any G-morphism ff: V ! U of bounded G-modules there is a G-morphism fi: W ! U of bounded G-modules such that fi ffi i = ff and kfik ~ kffk. Good candidates for relatively injective bounded G-modules are mapping spaces: Definition (2.16). Let G be a discrete group and let V be a Banach space. We write B(G, V ) for the real vector space of bounded functions G ! V . This vector 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. The supremum norm turns B(G, V ) into a bounded G-module. Moreover, this bounded G-module is relatively injective [Ivanov1 , Lemma (3.2.2)]. Definition (2.18). Let G be a discrete group. A G-resolution of a bounded G-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 a chain contraction (Kn)n2N consisting of (not necessarily equivariant) linear operators 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 norm and 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 2.3 Bounded cohomology of groups and (for n 2 N) dn :Vn ! Vn+1 i f 7 ! (g0, : :,:gn+1)(7! 1)n+1 f(g1, : :,:gn+1) n j + X ( 1)n j f(g0, : :,:gjgj+1, : :,:gn+1) j=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 "V v0 v1 0 V V0 V1 is a strong resolution of V and "U u0 u1 0 U U0 U1 is a complex of relatively injective bounded G-modules. Then any G-morphism w :V ! U can be extended to a G-morphism (wn)n2N of complexes: "V v0 v1 0 V V0 V1 w w0 w1 0 U "U U0 U1 . u0 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 defined in the same way as usual group cohomology: Definition (2.21). If G is a discrete group and V is a bounded G-module, we write V Gfor the set of G-fixed points. 37 2 Bounded Cohomology Definition (2.22). Let G be a discrete group and let v0 v1 R V0 V1 be a strong relatively injective resolution of the trivial bounded G-module R. Then the bounded cohomology of G (with coefficients in R), bH (G) , is the cohomology of the induced complex v0j G v1j 0 V0G V1 . Of course, one could also consider bounded cohomology of groups with more general coefficients [Monod ]. But in the following we will need only bounded cohomology with real coefficients. Corollary (2.23). Let G be a discrete group. Then any G-morphism between two strong relatively injective resolutions of the trivial bounded G-module R by bo* *unded G-modules extending the identity R ! R induces the same isomorphism on the cohomology of the G-fixed point sets. In particular, the bounded cohomology bH (G) is well-defined (up to canonical i* *so- morphisms). Proof.This follows directly from Theorem (2.20). A detailed proof is given __ by Ivanov [Ivanov1 , Section (3.7)]. |__| Each strong relatively injective resolution of the trivial G-module R gives rise to a seminorm on bH (G). The topology on bH (G) is independent of the chosen 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)], the so-called canonical seminorm. As in the case of ordinary group cohomology, each group homomorphism f :G ! H induces a homomorphism Hb (f): bH (H) ! Hb (G) (for a rigorous 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 classifying space of G. In fact, much more holds, as is explained in Section 2.4. 38 2.3 Bounded cohomology of groups 2.3.3 Amenability An interesting facet of bounded cohomology is that it ignores a large class of groups - the amenable groups. This gives rise to interesting applications to the simplicial volume (see Subsection 2.4.3). Definition (2.24). A group G is called amenable if the there is a G-invariant function m: B(G, R) ! R such that for all f 2 B(G, R) inff(g) ~ m(f) ~ sup f(g). g2G g2G Actually, this might seem a far-fetched definition. But amenable groups turn out 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 mean on G. Obviously, all finite groups are amenable - in this case m can be taken as integration over G with respect to the counting measure and dividing by the 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) ! Hb (G) induced by the canonical homomorphism G ! 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 resolution of R by bounded G-modules. Moreover, the induced canonical isomorphism turns out to be isometric. The theorem shows that the free group Z Z cannot be amenable: It can be shown 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 - a correct version can be found in the arcticles of Grigorchuk and Mitsumatsu [Grigorchuk ], [Mitsumatsu ]. 39 2 Bounded Cohomology 2.4 The mapping theorem _________________________________________________________________________ Perhaps surprisingly, the rather small differences between the definitions of singular and bounded cohomology lead to significant differences in the behaviour of both theories. For instance, Gromov and Ivanov showed that the bounded cohomology of spaces only depends on the fundamental group (Theorem (2.27)). More precisely, if f :X ! Y is a continuous map, there is a commutative diagram bH(f) bH (Y) Hb (X) ,= ,= bH (ss1(Y)) Hb (ss (X)) bH(ss1(f)) 1 whose vertical arrows are canonical isometric isomorphisms. This diagram allows us to understand the bounded cohomology of spaces via Ivanov's transparent 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 a space only depends on its fundamental group: Theorem (2.26). Let X be a simply connected CW-complex with countably many cells. Then the bounded cohomology of X is canonically isometrically isomorphic* * to the bounded cohomology of the one point space. Theorem (2.27). Let X be a connected countable CW-complex. Then bH (X) is canonically isometrically isomorphic to bH ss1(X) . Actually, these theorems remain true if we drop the condition that the spaces must be countable cellular. This situation is treated in Gromov's original exposition [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, we view bCk(X) as the set of bounded functions Sk(X) ! R. To prove the first theorem a contracting chain homotopy K0 K1 K2 R bC0(X) bC1(X) 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 Lane space K ssn(X), n which is a topological Abelian group Gn. Then there is such a tower p3 p2 p1 X3 X2 X1 = X consisting of principal Gn-bundles pn :Xn+1 ! Xn. Ivanov shows that in this situation for each n 2 N there is a chain map An :bC (Xn+1) ! bC (Xn) satisfying An ffi 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 show that (where p: eX ! X is the universal covering of X) 0 R Cb0(Xe) bC1(Xe) is a strong relatively injective resolution of the trivial bounded ss1(X)-mod- ule R. Here the left ss1(X)-action on bC(Xe) is given by the right ss1(X)-action on the universal covering space eX: for all f 2 bCk(X) and all g 2 ss1(X) g f := oe 7 ! f(x 7! oe(x) g) . Thus the bounded cohomology of the discrete group ss1(X) can be computed as the cohomology of the complex 0 Cb0(Xe)ss1(X) Cb1(Xe)ss1(X) The map bC (X) ! Cb eX induced by p establishes an isomorphism be- tween bC (X) and bC(Xe)ss1(X)commuting with the differentials. Hence the 41 2 Bounded Cohomology bounded cohomology of ss1(X) coincides with the cohomology of the com- plex @ 1 @ 0 Cb0(X) bC (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) ,=Hb (ffl). |__| 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 ss1(S1 ` S1) = Z Z, and bH2(Z Z)is non-trivial (see Subsection 2.3.3), but bH S1 ,=Hb (ffl). Example (2.30). There is no interesting theory of characteristic classes for real vector bundles in bounded cohomology: For each n 2 N, the fundamen- tal group of the classifying space BO(n) is amenable. Therefore the bounded cohomology bH (BO(n)) coincides with the bounded cohomology of the one point 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 and isomorphisms shows that Hb (f) = Hb (ss1(f))holds with respect to the isometric isomorphisms bH (X) ,= bH (ss1(X))and bH (Y) ,= bH (ss1(Y))of Theorem (2.27). Thus combining Theorem (2.27) and Theorem (2.25) yields: Theorem (2.31) (Mapping Theorem for Bounded Cohomology). Let X and Y be countable CW-complexes and let f :X ! Y be a continuous map. If the homomorphism ss1(f): ss1(X) ! ss1(Y) is surjective and its kernel is amenable, then the induced homomorphism bH (f)= bH (ss1(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 homomorphism H (f, R)then also preserves the seminorm in homology. Again, the mapping theorem remains true if we drop the condition for the spaces 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 following results 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, ss1(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 fl fl kMk = fl[M]Rfl1= 0, __ by duality (the first part of Theorem (2.7)). |__| A nice argument from covering theory yields the following result [Lu"ck2, Exercise 14.11]: Corollary (2.33). Let M and N be oriented closed connected manifolds of the same dimension and let f :M ! N be a continuous map of degree 1. Then ss1(f) is surjective. If the kernel of ss1(f) is amenable, then kMk = kNk . Example (2.34). However, the simplicial volume does not only depend on the 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 fundamental group, but kF2k = 4 and kF2 S2k = 0 by Example (1.24) and Theorem (2.14) respectively. 43 2 Bounded Cohomology Furthermore, one can derive from the mapping theorem that the simplicial volume is additive with respect to connected sums. Theorem (2.35). Let M and N be two oriented closed connected manifolds of the same dimension n ~ 3. Then k M #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 and Bounded 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 ingredients needed 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. o Then ss1(X) acts from the left on the vector space map (Xek+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 (Xek+1, R) and all g 2 ss1(X). o The subset of bounded functions in map (Xek+1, R) is denoted by F (X) and we use the abbreviation Ik(X)for the functions in Fk(X) which are invariant under the above ss1(X)-action. How can we turn F (X) into a cochain complex? The vector space bCk(Xe) can be identified with the space of bounded functions Sk(Xe) ! R (under this 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(Xe), namely as the space of those bounded functions Sk(Xe) ! R which only depend on the vertices of the simplices (and are continuous in the vertices). In Gromov's terminology those functions would be called "straight bounded continuous cochains" [Gromov2 , Section 2.3]. It is clear that the coboundary operator on bC(Xe) restricts to F (X) and that the operations of ss1(X) on F (X) and bC(Xe) are compatible with the above inclusion map. This makes F (X) a subcomplex of bC(Xe). In other words, the homomorphism uk: Fk(X) ! bCk(Xe) f 7 ! oe 7! f(oe(e0), : :,:oe(ek)) is a ss1(X)-equivariant cochain map. Theorem (2.37). Let X be a connected countable CW-complex and let eX be its universal covering space. 1. Then the homomorphism ": R ! F0(X) = map (Xe, R) c 7 ! (x 7! c) turns the cochain complex F (X) into a strong relatively injective resolut* *ion by bounded ss1(X)-modules of the trivial ss1(X)-module R. 2. Hence the cohomology of I (X) is canonically isomorphic to Hb (ss1(X)). Moreover, under this isomorphism the supremum norm on F (X) induces the canonical seminorm on bH (ss1(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 description of this map in Section 4.3, we will now construct such a cochain map with the help of the uniqueness part of Theorem (2.20). 45 2 Bounded Cohomology As mentioned in Section 2.4, the bounded cohomology groups bH (ss1(X)) can be computed via the strong relatively injective resolution bC(Xe). There- fore we will compare the two complexes I (X) and bC(Xe): The "inclusion" map u from above fits into the ladder " 0 1 0 R F (X) F (X) id u0 u1 0 R " Cb0(Xe) bC1(Xe) . Now Corollary (2.23) implies that (after taking the ss1(X)-fixed points) u induces the canonical isomorphism ss (X) H I (X) ! H Cb (Xe) 1 (which is isometric by Theorem (2.37)). As mentioned in Section 2.4, the isometric isomorphism ss (X) bp:Hb (X) ! H Cb (Xe) 1 is induced by the universal covering map p: eX ! X. Thus the desired iso- metric isomorphism H I (X) ! Hb (X) is the homomorphism H (w ) making the triangle H (w ) H I (X) bH (X) H (uj ) bp H Cb (Xe)ss1(X) commutative (where uj denotes the restriction of u to the ss1(X)-fixed points). The inverse of bpon the level of cochains can be described as follows: for all f 2 bCk(Xe)ss1(X)and oe 2 Sk(X) we define k v (f) (oe) := f(eoe), where eoeis some lift of oe with respect to p. This definition does not depend on the chosen lift since f is ss1(X)-invariant and any two lifts of oe only dif* *fer by multiplication with an element in ss1(X) [Massey , Lemma V 8.1]. Since f is bounded, vk(f) is bounded on Sk(X) . Therefore linear extension yields vk(f) 2 bCk(X). 46 2.5 A special resolution Covering theory implies that for all oe 2 Sk(X) and all j 2 f0, : :,:kg each li* *ft of the j-th face of oe is also the j-th face of some lift of oe. Hence v : bC(Xe)ss1(X) ! bC (X) is a cochain map. By construction, it is the inverse of the cochain map bC (X) ! bC(Xe)ss1(X)induced by the universal covering map p. Therefore, the composition w := v ffi uj : I (X) ! bC (X) induces the canonical isometric isomorphism H (I (X) ) ! Hb (X). By definition, for all f 2 Ik(X) and all oe 2 Sk(X) we obtain k w (f) (oe) = f eoe(e0), : :,:eoe(ek) , where eoeis any lift of oe. This is the description we will use in Section 4.3. 47 2 Bounded Cohomology 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 spa* *ce of singular simplices whose mass is concentrated in finitely many points. In the measure 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 homology should coincide. More extensive accounts are the papers of Zastrow [Zastrow ] a* *nd Hansen [Hansen ] and the book of Ratcliffe [Ratcliffe]. The motivation for the introduction of measure homology originates from the fact that measure homology can be used to calculate the simplicial volume, hence giv- ing more room for constructions such as smearing (cf. Chapter 5). This smearing construction will be the key to the proof of the proportionality principle. In Section 3.1, the technical background needed for the definition of measure h* *o- mology is presented - in particular, a feasible topology on the set of smooth s* *in- 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 i* *s also given a proof for the compatibility of measure homology with colimits. The next chapter is devoted to the proof that measure homology and singular homology are isometrically 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. In the later subsections, topologies for mapping spaces are provided and some technicalities concerning smooth maps on standard simplices are discussed. 3.1.1 Signed measures Signed measures are a slight generalisation of positive measures. As the name suggests, signed measures can also take negative values. The gener- alisation is quite straightforward; however, some additional care has to be taken 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 finite signed measures on a given measurable space is a vector space over the real numbers. Definition (3.1). Let (X, A) be a measurable space. o 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 oe-additive. o A null set of a signed measure ~ is a measurable set A with ~(B) = 0 for all B 2 A with B ae A. o A set A 2 A is called ~-positive if ~(B) ~ 0 for all B 2 A with B ae A. Analogously, ~-negative sets are defined. o A determination set of ~ is a subset D of X such that each measurable set 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 since each 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 = P0[ N0 is another decomposition in a ~-positive and a ~-negative set, then the symmetric difference of P and P0 (which equals the symmetric difference of N and N0) is a ~-null set. A proof is, for example, given in Elstrodt's textbook on measure theory and integration [Elstrodt, Satz 1.8 in Kapitel VII]. Definition (3.3). Let ~ be a signed measure on (X, A) and X = P [ N its Hahn decomposition. o 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 ~. o 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 the negative variation of a signed measure are indeed well-defined. Moreover, the sum ~+ + ~ makes sense because at least one of these two measures must 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 = sup ~(A) inf ~(B). A2A B2A 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 oe-algebra is the oe-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 a measurable function (with respect to the Borel oe-algebra on R). If ~ is a signed 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 Z Z f d~ := f d~+ f d~ . Lemma (3.7). Let X be a topological space. The set of all signed measures on the Borel oe-algebra of X possessing a compact determination set and finite total v* *aria- tion is a real vector space. The total variation turns this vector space into a* * normed vector space. Proof.Let ~ and be two such measures and let ff 2 R. Obviously, ff ~ again is a signed measure. Each determination set of ~ is a determination set of ff ~ and kff ~k= jffj k~k . Since both ~ and have finite total variation, their value on each Borel set is 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 oe-algebra on X) k~ + k = sup ~(A) + (A) inf ~(B) + (B) A2A B2A ~ sup ~(A) + sup (A) inf ~(B) inf (B) A2A A2A B2A B2A = k~k + k k. __ In particular, k~ + k is finite. |__| Definition (3.8). Let f :X ! Y be a Borel function, i.e., measurable with respect to the Borel oe-algebra. Moreover, let ~ be a signed measure on the Borel oe-algebra of X. Then ~ induces a signed measure ~f on Y given by 1 ~f(A) := ~ f (A) for all Borel subsets A ae Y. Lemma (3.9). Let f :X ! Y be a continuous function and let ~ be a signed mea- sure on the Borel oe-algebra of X possessing a compact determination set and fi* *nite total variation. Then the induced measure ~f also has a compact determination s* *et and finite total variation. Proof.It is easy to see that continuous functions are Borel functions since the Borel oe-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 a determination set of ~f. Since f is continuous, f(D) is compact. Writing AX and AY for the Borel oe-algebras on X and Y respectively we compute the total variation of ~f via Lemma (3.4) as follows fl fl fl~ffl= sup ~f(A) inf ~f(B) A2AY B2AY 1 1 = sup ~ f (A) inf ~ f (B) A2AY B2AY ~ sup ~(A) inf ~(B) A2AX B2AX = 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 set of singular simplices, we have to introduce an appropriate topology on the corresponding mapping space. Unfortunately, the compact open topology is too coarse for our purposes. So we need another topology - the so-called C1-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) the set of continuous maps X ! Y. o If K ae X is compact and U ae Y is open, we write UK := ff 2 map (X, Y)j f(K) ae Ug. o The compact open topology on the mapping space map (X, Y)is the topology with the subbase fUK j K ae X compact, U ae Y openg. Remark (3.11). In particular, for all x 2 X the evaluation map map (X, Y) ! Y f7 ! f(x) __ is continuous. |__| Definition (3.12). Let M and N be two smooth manifolds without bound- ary. As sets map 1 (M, N) and map co1(M, N) are just the sets of smooth maps M ! N. o The topology on map co1(M, N)ae map (M, N) is the subspace topology given by the compact open topology on map (M, N) . o The topology on map 1 (M, N) , called C1-topology, is the topology consisting of all inverse images of the form T 1 (U) where U is an open subset of map (TM, TN) and T is the differential: T :map 1 (M, N) ! map (TM, TN) f 7 ! Tf. I.e., the injective map T :map 1 (M, N) ! map (TM, TN) is turned into a homeomorphism onto its image T map 1 (M, N) . 54 3.1 Prelude More information about the C1-topology and similar topologies can be found in Hirsch's book [Hirsch1 , Chapter 2]. The above description is taken from a paper of Hirsch [Hirsch2 , page 244]. Remark (3.13). The compact open topology is the same as the topology of uniform convergence on compact subsets [Dugundji , Theorem XII 7.2]. Sim- ilarly, the C1-topology is the topology of uniform convergence on compact subsets not only of the function itself but also of its differential. Lemma (3.14). Let M and N be two smooth manifolds without boundary. Then the C1-topology on map 1 (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 ss, we get for all compact sets K ae M and all open sets U ae N that K T U " map 1 (M, N) fi K = Tf :TM ! TN fif 2 U " map 1 (M, N) fi 1 = Tf :TM ! TN fif 2 map 1 (M, N) , Tf(i(K)) ae ss (U) fi 1 = g 2 map (TM, TN) fig(i(K)) ae ss (U) " T map 1 (M, N) . Since i and ss are continuous, i(K) ae TM is compact and ss 1 (U) ae TN is__ open. Thus the set UK " map 1 (M, N) is open in map 1 (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 ! h ffi f and map (X, Y) ! map (X, Z) h 7 ! g ffi h are continuous. 2. Let L, M, N be smooth manifolds without boundary and f 2 map 1 (L, M), g 2 map 1 (M, N) . Then the maps map 1 (M, N) ! map 1 (L, N) h 7 ! h ffi f 55 3 Measure Homology and map 1 (L, M) ! map 1 (L, N) h 7 ! g ffi h are continuous. Proof.1. This can easily be seen using the definition of the compact open topology. 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 taken when defining the set of all smooth maps k ! M, where M is some smooth manifold without boundary. The elements of map 1 ( k, M) should be 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. o The k-dimensional hyperplane in Rk+1 spanned by the vertices of the standard k-simplex k will be denoted by k. We then write k for the set of all open sets V ae k (in the subspace topology) containing k. o A map f : k ! M is smooth if there is a V 2 k and a smooth map F 2 map 1 (V , M)such that Fj k = f. The pair (F, V ) is called an extension of f. o A map f :M ! k is called smooth if the composition i ffi 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 a standard simplex and f is smooth in the above sense. o We write map 1 ( k, M) and map 1 (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 then we obviously get an inclusion map 1 ( 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 map co1( k, M) for the set map 1 ( k, M) ae map ( k, M) endowed with the subspace topology. However, we will also need some kind of C1-topology on map 1 ( k, M). So we introduce the differential of a smooth map k ! M: Definition (3.19). For k 2 N we 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 map 1 ( k, M) we define its differential by Tf := TFjT 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 map 1( k, M). Suppose (F1, V1) and (F2, V2) are two extensions of f. Then TF1jss 1 k= TF2j 1 k 1 ( ) ss2 ( ) where ss1: TV1 ! V1 and ss2: TV2 ! V2 are the bundle projections. Proof.Since F1 and F2 are smooth, their differentials TF1 and TF2 are contin- uous. Furthermore, TF1jss 1 k ffi= TF2j 1 k ffi 1 (( ) ) ss2 (( ) ) 57 3 Measure Homology since F1 and F2 extend f. For each (x, v) 2 ss11 ( k) = ss21 ( k) there is a sequence (xn)n2N ae ( k)ffisuch that x = lim xn, n!1 which implies limn!1 (xn, v) = (x, v) in ss11 ( k) = ss21 ( k). Hence we conclude TF1(x, v)= TF1 lim(xn, v) = lim TF1(xn, v) n!1 n!1 = lim TF2(xn, v) = TF2 lim(xn, v) n!1 n!1 __ = TF2(x, v). |__| Having defined the differential of a smooth map k ! M, we are now able to introduce the C1-topology on map 1 ( 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 map 1 ( k, M) is the topol- ogy induced by the compact open topology on map (T k, TM) pulled back by the differential (as defined in Definition (3.20)) k k T :map 1 , M ! map T , 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 map 1 ( k, M) is finer than the topology of map co1( 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 to the 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 ae M an open subset. Let k 2 N. 1. Then the map j :map 1 k, N ! map 1 k, M , induced by the inclusion N ,! M, is continuous, injective, and open. 58 3.1 Prelude 2. Let im j := j map 1( k, N) ae map 1 ( k, M) (endowed with the subspace topology). Then the inverse im j ! map 1 ( k, N) of j is continuous. Proof.1. Since the inclusion N ,! M is smooth, continuity of j follows from Remark (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 is compatible with intersections. Hence it suffices to show that j maps sets of the form T 1 (UK ) with open U ae TN and compact K ae T k to open sets in map 1 ( k, M). By definition, 1 K j T (U ) k fi = j f 2 map 1 ( , N) fiTf(K) ae U fi k = i ffi f fif 2 map 1 ( , N), Tf(K) ae U k fi k fi k = f 2 map 1 ( , M) fiTf(K) ae U " f 2 map ( , M) fif( ) ae N where i :N ,! M is the inclusion. Since N is open in M, the set U must be open in TM. Thus the first set is open in map 1 ( k, M). Lemma (3.23) shows that the second set is also open in map 1 ( 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)k2N be an ascending family of open sets covering M. Let j 2 N. For each k 2 N the map map 1 ( j, Uk) ! map 1( j, M) induced by the inclusion Uk ,! M is denoted by jk. Then jk(map 1( j, Uk)) k2N is an ascending open covering of map 1 ( j, M). Proof.It is clear that this family is ascending. By Lemma (3.25) all the im- ages jk map 1 ( j, Uk) are open in map 1 ( j, M). Moreover, map 1( j, M) is covered by jk(map 1( j, Uk)) k2N : namely, if f 2 map 1 ( j, M), then the image f( j) ae M is compact. Since (Uk)k2N is an ascending open covering of M, there is some k 2 N such that f( j) ae 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 FjV 0, V 0 shows that there is an f 2 map 1 ( 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 ae map k, X be some set of simplices. The idea of measure homology is to think of a singu- lar chain P oe2Saoe oe with real coefficients as a signed measure on S having the mass aoeon the set foeg. The measure chain complex will consist of all signed measures on S (modulo some finiteness condition). In particular, cer- tain "infinite" singular chains are allowed. Thus the measure chain complex is 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 the coin is that it is quite hard to get a geometric intuition of more complicated measure chains. Depending on the choice of the mapping space S (and its topology) there are two main flavours of measure homology: o One for general (pairs of) spaces using the compact open topology on the set of all singular simplices, and o one for (pairs of) smooth manifolds using the C1-topology on the set of smooth singular simplices. However, our application of measure homology in Chapter 5 compels us to use the smooth version. So we will only introduce this version in detail. Measure homology was introduced by Thurston [Thurston , page 6.6]. Some basic properties of smooth measure homology are provided in Ratcliffe's book [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 refer to those adapted statements of his paper by numbers such as, e.g., 4.5diffin- stead of 4.5. Unfortunately, it is probably more transparent to provide a new proof 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 this chapter. 60 3.2 Definition of measure homology Definition (3.27). Let M be a smooth manifold without boundary and k 2 N. o The k-th measure chain group, denoted by Ck(M) , is the R-vector space of all signed measures on the Borel oe-algebra of the mapping space map 1 ( k, M) possessing a compact determination set and finite total variation. Its elements are called measure k-chains. o 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 (which we will also denote by @j) @j:Ck+1 (M) ! Ck(M) ~ 7 ! ~(oe7!oeffi@j) We then define the boundary operator of measure chains by @: Ck+1 (M) ! Ck(M) k+1 ~ 7 ! X ( 1)j @j(~). j=0 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 r^ole of the compactness of the standard simplex (as can be seen in Lemma (3.45)) and the finiteness of singular chains in 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_1_ in x11.5]. |__| Statements like the previous lemma are proved by Zastrow, using a more general framework which permits to translate some properties of singular homology 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 ker @: Ck(M) ! Ck 1 (M) Hk(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 let k 2 N. The total variation k kinduces a seminorm on Hk(M) as follows: For all ~ 2 Ck(M) we define fi k~k mh := inf k k fi 2 Ck(M) , @( ) = 0, [ ] = ~. Just like in singular homology, there is also a notion of relative homology groups: Definition (3.31). Let M be a smooth manifold without boundary and let N be a smooth submanifold without boundary. Due to Remark (3.24) and Lemma (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 can define the k-th relative measure chain group as the quotient vector space Ck(M) Ck(M, N) := _______. Ck(N) The boundary operator on the absolute measure chain groups clearly in- duces a boundary operator on the relative measure chain groups, turning them into a chain complex. Definition (3.32). Let M be a smooth manifold without boundary and let N ae M be a smooth submanifold without boundary. For each k 2 N the k-th relative measure homology group is defined to be the quotient ker @: Ck(M, N) ! Ck 1 (M, N) Hk(M, N) := __________________________________. im @: Ck+1 (M, N) ! Ck(M, N) Remark (3.33). Let M be a smooth manifold without boundary. Since there is 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 maps measure homology behaves like singular homology with real coefficients; actually, most of the proofs are based on the same ideas as the corresponding proofs for singular homology theory. Lemma (3.34) (Dimension Axiom). Measure homology satisfies the dimension axiom: For all n 2 N ( R ifn = 0, Hn (ffl),= 0 otherwise. Proof.It is easy to see that the chain complexes C (ffl), @ and C (ffl,,R)@ are the same because for each k 2 N there is just one map k ! ffl (which is smooth). Thus H (ffl),=H (ffl, R) __ which proves the lemma. |__| Lemma (3.35) (Naturality). Each smooth map f :M ! N between smooth manifolds 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 k k map 1 , M ! map 1 , N oe7 ! f ffi oe, which is continuous by Remark (3.24). Hence Lemma (3.9) shows that we obtain an induced map Ck(M) ! Ck(N) ~ 7 ! ~(oe7!fffioe) which obviously is a homomorphism. This yields a chain map C (M) ! C (N) [Zastrow , Lemma 2.10diff]. Hence we get an induced homomorphism __ H (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 for measure homology, i.e., if M is a smooth manifold without boundary and N ae M is a smooth submanifold without boundary, then the sequence Hk(i) Hk(j) Hk+1 (M, N) Hk(N) Hk(M) 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) is natural in M and N. Proof.The definition of relative measure chain groups yields a short exact sequence 0 Ck(N) Ck(M) Ck(M, N) 0 , where the maps are induced by the (smooth) inclusions i and j. Therefore we 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 the above theorem to be smoothly homotopic. But smooth homotopic maps are always 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 of the excision axiom: Let M be a smooth manifold without boundary and let W ae M be a smooth submanifold without boundary. If U ae W is a subset such that M n U* * is a smooth manifold without boundary and W n U ae M n U is a smooth submanifold without boundary and if there is some subset V ae M such that __ ffi __ U ae V ae V ae W, then the inclusion (M n U, W n U) ,! (M, W) induces an isomorphism H (M n U, W n U) ,=H (M, W) . 64 3.3 Basic properties of measure homology Definition (3.40). A topological space X is called normal if disjoint closed sets A, B ae X can be separated by open sets, i.e., there are disjoint open sets U, V ae X with A ae U and B ae V . Remark (3.41). All metric spaces are normal. In particular, every smooth manifold is normal (since every smooth manifold can be given a Riemannian structure, and thus yielding a metric). Corollary (3.42). Let M be a smooth manifold without boundary and W ae M an open subset. If U ae M is a closed subset with U ae W, then the inclusion (M n U, W n U) ,! (M, W) induces an isomorphism H (M n U, W n U) ,=H (M, W) . Proof.Since W ae M is open, W is a smooth submanifold without boundary. Moreover, M n U is a smooth manifold without boundary and the subset W n U ae M n U is a smooth submanifold without boundary because U is closed. Thus it remains to show that there is an open subset V ae M with __ U ae V ae V ae W. Since M is normal and U ae W, we can separate the disjoint closed sets U and M n W by open sets V and V 0. In particular, __ 0 U ae V ae V ae M n V ae W. __ Hence excision proves the corollary. |__| Corollary (3.43) (Mayer-Vietoris Sequence). Suppose M is a smooth manifold without boundary and U, V ae 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 an isomorphism 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) and the arrow on the right is the connecting homomorphism of the long exact measure 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 an ascending family (Uk)k2N of open subsets. Then the inclusions Uk ,! M induce an isomorphism colim H (Uk),= H (M) . k!1 The idea of the proof is essentially the same as in singular homology: to use the compactness of the standard simplex. However, we have to be careful about 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 3.3 Basic properties of measure homology where jk: map 1 ( j, Uk) ! map 1( 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 open ascending covering of map 1 ( j, M), using the notation j 8k2N im jk := jk map 1 ( , Uk) . Let D be a compact determination set for ~. Hence there is an index k 2 N such that D ae jk map 1 ( j, Uk) . Let Ak be the Borel oe-algebra of im jk (with the subspace topology). By Lemma (3.25) the set im jk is open in map 1 ( j, M). Thus Ak is contained in the Borel oe-algebra of map 1 ( j, M). Therefore we may define the signed measure e~on im jk via 8A2Ak e~(A) := ~(A). By construction, D ae im jk, implying that D is a compact determination set of e~. Using Lemma (3.4), it is easy to see that e~has finite total variation. If Jk: im jk ! map 1 ( 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 map 1 ( 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 ae map 1 ( j, M) be some Borel set. Hence eA:= A " im jk 2 Ak and ~(A) = ~(Ae) + ~ A " (map 1( j, M) n im jk) . Since map 1 ( j, M) n im jk is a measurable set contained in the complement of the determination set D, ~(A) = ~(Ae) holds. Therefore the constructions of jk and Jk yield ~(A) = ~(Ae) = e~(Ae) 1 1 J 1 = e~Jk jk (Ae) = e~ kjk (A " im jk) 1 1 = jk (A " im jk) = jk (A) = jk(A). 67 3 Measure Homology Since jk is induced by the (smooth) inclusion Uk ,! M, the map jkis a chain map. In particular, @~ = @( jk) = (@ )jk. Suppose ~ is a cycle and A ae map 1( j, Uk) is a Borel set. Then jk(A) = Jk1 (A) lies in Ak. As mentioned above, this results in jk(A) being Borel in map 1 ( j, M). Using the injectivity of jk we therefore compute 1 j @ (A) = @ jk (jk(A)) = (@ ) k 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 system with respect to the structure maps Hj(i`k) (k,`)2Dwhere 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: colim Hj(Uk) ! Hj(M) k!1 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 to show that for each ff 2 Hj(M) there exists a k 2 N and a fi 2 Hj(Uk) such that Hj(ik)fi = ff, and that for each ff 2 Hj(Uk) with Hj(ik)(ff) = 0 there is an ` 2 N such that Hj i`k(ff) = 0. 1. Let ~ 2 Cj(M) be a measure cycle. Then the preceding lemma provides us 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 o 2 Cj+1(M) with @o = jk. 68 3.3 Basic properties of measure homology The above lemma gives us an ` 2 N (and the proof shows that we can assume ` ~ k) and a measure chain % 2 Cj+1(U`) satisfying %j`= o, (@%)j`= @o = jk. Writing j`k:map 1 ( j, Uk) ! map 1( j, U`) for the map induced by the inclusion i`k, we conclude j` j` (@%)j`= jk= k . As in the last part of the proof of Lemma (3.45), we can compute for all Borel sets A ae map 1 ( j, Uk) 1 j @%(A) = @% j` (j`(A)) = (@%) ` j`(A) j` j` j` 1 = k j`(A) = k j` (j`(A)) ` = jk(A). __ This shows Hj i`k [ ] = 0 in Hj(U`) . |__| With the help of these algebraic properties we will see in Section 4.2 that measure homology and singular homology (algebraically) coincide. 69 3 Measure Homology 70 Measure Homology is 4 Singular Homology _________________________________________________________________________ In this chapter, singular homology (with real coefficients) and measure homology of smooth manifolds without boundary are compared. The results from Section 3.3 already suggest that the homology groups of these two theories should coincide. Surprisingly, the natural isomorphism providing this correspondence turns out to be isometric. It is thus possible to calculate the simplicial volume of an ori* *ented smooth 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: oFirst of all, we introduce smooth singular homology (for smooth manifolds without boundary) and show that it is canonically isometrically isomorphic to singular homology. oIn the second step, we establish an (algebraic) isomorphism between smooth singular homology and measure homology. This is done by means of the properties described in Section 3.3. oThirdly, we prove that this isomorphism is actually isometric. This is the central contribution of this thesis. The method of proof is inspired by t* *he duality principle of bounded cohomology (Theorem (2.7)). In the last section, we discuss an integration operation for measure chains as * *a tool to analyse the top homology group, which will turn out to be useful in the cont* *ext of 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 see a connection between measure homology and smooth singular homology instead of singular homology. Smooth singular homology is defined just like ordinary singular homology, using smooth singular chains instead of continuous ones. Definition (4.1). Let M be a smooth manifold without boundary and k 2 N. The elements of map 1 ( k, M) are called smooth singular k-simplices of M. The subcomplex (using the inclusion from Remark (3.17)) sm C (M, R), @jCsm(M,R) ae C (M, R), @ generated by all smooth singular simplices yields the smooth singular ho- mology groups ker @: Csmk(M, R) ! Csmk 1(M, R) Hsmk(M, R) := ___________________________________ im @: Csmk+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]. In Massey's book it is shown (using the same methods as in ordinary singular homology) that smooth singular homology satisfies theorems correspond- ing to those stated in Section 3.3. These statements will in the following be referred to via numbers such as (3.39)sminstead of (3.39). Moreover, by the same pattern as for singular homology, we can define a norm k ksm1on the smooth singular chain complex and its induced semi- norm k ksm1on smooth singular homology. Theorem (4.2). For every smooth manifold M without boundary the natural chain complex inclusion jM : Csm (M, R) ,! C (M, R) induces an isometric isomorphism between the homology groups Hsm (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: o The map s is a chain map with s ffi jM = id and jM ffi s ' id. o For each singular simplex oe 2 Sk(M) the image s(oe) 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 oe2Sk(M)aoe oe 2 Ck(M, R) is a cycle representing the homology class ff 2 Hk(M, R) , then P oe2Sk(M)aoe s(oe) is a cycle representing Hk(s)(ff) with fl flsm fl fl kHk(s)(ff)ksm1~ flflX aoe s(oe)flfl~ X jaoej oe2Sk(M) 1 oe2Sk(M) fl fl fl fl = flflX aoe oeflfl. oe2Sk(M) 1 Taking the infimum over all representatives of ff thus yields fl fl k Hk(s)(ff)ksm1~ kffk1= flHk(jM ) Hk(s)(ff) fl1. On the other hand, obviously fl fl fl sflm flHk(jM )(ff0)fl~ flff0fl 1 1 holds for all ff0 2 Hsmk(M, R). Since Hk(s) is a bijection this proves the_ theorem. |__| In particular, we detect the fundamental class in smooth singular homology and we can compute the simplicial volume via this class: Definition (4.3). Let M be an oriented smooth closed connected manifold of dimension n. The above theorem implies that 1 [M]sm := Hn(jM ) [M]R is a well-defined generator of Hsmn(M, R). 73 4 Measure Homology is Singular Homology Corollary (4.4). Let M be an oriented smooth closed connected manifold. Then fl flsm fl[M]smfl = kMk 1 holds by the above theorem. In Section 4.3, we will make use of a version of bounded cohomology based on smooth singular homology. Its construction is completely analogous to the corresponding definition of bounded (singular) cohomology and it turns out to be the same as ordinary bounded cohomology. However, bounded smooth 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. o The supremum norm of f 2 Hom R Csmk(M, R), R is given by kfk 1 := sup jf(oe)j. oe2map1 ( k,M) o We write fi bCksm(M) := f 2 Hom R Csmk(M, R), R fikfk1 < 1 for the group of bounded smooth k-cochains. o The corresponding coboundary operator is defined via ffi :bCksm(M)! bCk+1sm(M) k+1 f 7 ! c 7! ( 1) f(@(c)) . o The k-th bounded smooth cohomology group Hbksm(M) of M is the k-th cohomology group of the cochain complex (Cbsm(M) , ffi). o The supremum norm induces (as in the case of ordinary bounded co- homology) a seminorm on the quotient bHksm(M), which will also be denoted by k k1. Remark (4.6). As in the case of ordinary bounded cohomology, it is easy to * * __ verify that the homomorphism ffi is well-defined and satisfies ffi ffi ffi = 0.* * |__| Using the smoothing operator s of the proof of Theorem (4.2), it can readily be seen that bHsm(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)) Cb (jM ):bC (M) ! bCsm(M) f 7 ! fjCsm(M,R)= f ffi jM induces an isometric isomorphism bH (jM ):bH (M) ! Hbsm (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 ffi s ' id. The same arguments as in the proof of Lemma (2.5) yield that the "dual" Cb (M) ! bC 1(M) f 7 ! ( 1) f ffi h 1 is well-defined and a cochain homotopy bC (s)ffi bC (jM )' id, where bC (s) and bC (jM )are the "duals" of s and jM respectively. Moreover, Cb (jM )ffi 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 k f ffi1sk~ kfk1 for all f 2 bCsm(M) . On the other hand, obviously kf ffi jM1k~ kfk1 holds for all f 2 bC (M) . Therefore we conclude fl fl fl fl 8'2Hb(M) k'k 1 = flbH(s) Hb (jM )(') fl1 ~ flbH(jM )(')fl1 ~ k'k 1, __ which implies that bH (jM )is isometric. |__| As in the case of ordinary bounded cohomology, we can define a Kronecker product h , : ibCsm(M) Csm (M, R) ! R. by evaluation. Analogously, this product descends to a bilinear map h , :Hibsm(M) Hsm (M, R) ! R, 75 4 Measure Homology is Singular Homology and it is readily seen to be compatible with the ordinary Kronecker product, i.e., for all ' 2 bH (M) and all ff 2 Hsm (M, R) ff ff bH (jM )('), ff = ', H (jM )(ff) . 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). Let M be a smooth manifold without boundary and let k 2 N and ff 2 Hsmk(M, R). 1. Then kffksm1= 0 if and only if 8'2Hbksm(M) h', ffi= 0. 2. If kffksm1> 0, then n 1 fi o kffksm1= sup ______fifi' 2 bHksm(M), h', ffi= 1 . k'k 1 Proof.By the above remark, k k 1 ff h', ffi= bH (jM )ffi bH (jM ) ('), ff k 1 ff = bH (jM ) ('), Hk(jM )(ff) holds for all ff 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 duality __ principle (2.7) applied to Hk(jM )(ff) 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 a norm-preserving injective chain map iM : Csm (M, R) ! C (M) . 76 4.2 The algebraic isomorphism Proof.Let k 2 N. For oe 2 map 1( k, M) we denote the atomic measure on map 1 ( k, M) concentrated in oe by ffioe, which is obviously an element of Ck(M) . If c := P oe2map1 ( k,M)aoe oe is a smooth chain, we define iM (c) := X aoe ffioe2 Ck(M) . oe2map1 ( k,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 we obtain that the Hahn decomposition of iM (c) is given by k fi P := oe 2 map 1 ( , M) fiaoe~ 0 , k fi N := oe 2 map 1 ( , M) fiaoe< 0 . Hence we conclude k kiM (c)k= jiM (c)j map 1( , M) = iM (c)(P) iM (c)(N) = X aoe X aoe oe2P oe2N = X jaoej oe2map1 ( k,M) __ = kcksm1. |__| Theorem (4.10). If M is a smooth manifold without boundary, then the inclusion iM : 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 and Zastrow via the verification of the Eilenberg-Steenrod axioms [Hansen , The- orem 1.1], [Zastrow , Theorem 3.4]. In the smooth setting however, it is not entirely clear how to use this approach since only smooth manifolds can be handled (instead of the whole universe of CW-complexes). Therefore we give a proof based on some "manifold induction" - following a general idea of Milnor: Theorem (4.11). Assume P is some property of smooth manifolds without bound- ary satisfying the following conditions: 77 4 Measure Homology is Singular Homology 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 a zero-dimensional smooth manifold without boundary). 3. If U, V are open subsets of the same smooth manifold without boundary such that 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 someS(common) smooth manifold without boundary all having property P, then k2NUk satisfies P. Then each smooth manifold without boundary has property P. Proof.The (not very difficult) proof can for example be found in Massey's__ book [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 the homomorphism 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 mentioned properties: 1. Suppose f :M ! N is a smooth homotopy equivalence of smooth man- ifolds without boundary where M enjoys P. Then the diagram Hsm(f,R) sm Hsm (M, R) H (N, R) H (iM ) H (iN) H (M) H H (N) (f) 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 (ffl)and Csm (ffl, R)are identical and the inclusion iffl:C (ffl),! Csm (ffl, R)is the identity. Hence the one point space satisfie* *s P. 78 4.2 The algebraic isomorphism 3. Let U, V be open subsets of some smooth manifold M without boundary such that U, V , and U " V satisfy P. Then we obtain the following commu- tative ladder Hsmk+1(U, R) Hsmk(U, R) Hsmk+1(U " V , R) Hsmk+1(U [ V , R)Hsmk(U " V , R) Hsmk+1(V , R) Hsmk(V , R) Hk+1(iU"V),= Hk+1(iu),H= Hk+1(iU[V) Hk(iU"V),=Hk(iU) Hk(iV),= k+1(iV) Hk+1(U) Hk(U) Hk+1(U " V ) Hk+1(U [ V ) Hk(U " V ) Hk+1(V ) 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 [ V has property P. 4. Let M be a smooth manifold without boundary and (Uk)k2N an ascending sequence of open subsets of M. Suppose that each Uk satisfies P. S Since 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) and Theorem (3.44)sm that the inclusions induce isomorphisms i :colim H (Uk) ! H (U), k!1 j :colim Hsm (Uk, R) ! Hsm (U, R) k!1 using the notation [ U := Uk. k2N Furthermore, for each k 2 N the diagram Hsm(Uk,!U,R)sm Hsm (Uk, R) H (U, R) H (iUk) H (iU) H (Uk) H H (U) (Uk,!U) commutes. Since each Uk has property P, we conclude that the induced homomorphism colim H (iUk): colim Hsm (Uk, R) ! colim H (Uk) k!1 k!1 k!1 79 4 Measure Homology is Singular Homology is an isomorphism. The previous diagram shows that j sm colimk!1 Hsm (Uk, R) H (U, R) colimk!1H (iUk) H (iU) colimk!1 H (Uk) i H (U) __ is commutative, implying that H (iU ) is also an isomorphism. |__| In particular, we obtain a fundamental class for measure homology. It is the aim of this chapter to show that this fundamental class can be used to compute the simplicial volume. Definition (4.12). Let M be a smooth oriented closed connected manifold of dimension 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 be true [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 appeared to remedy this situation [Bowen ], but the given argument does not seem to be 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 write jM : Csm (M, R) ,! C (M, R) for the obvious inclusion of chain complexes and iM : Csm (M, R) ! C (M) for the injective norm-preserving chain map which is given by mapping smooth singular simplices to the corresponding atomic 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 inclusion iM : 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., fl fl __ kMk = fl[M]mhflmh. |__| 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 the general 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 open topology 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 results of Theorem (2.7) (and Corollary (4.8)). This yields a reformulation of the problem in terms of bounded cohomology, which can be solved by means of the established framework sketched in Section 2.3. The drawback of this approach is that the proof of Theorem (4.14) is not as geometric 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 r^ole of the bounded (smooth) cohomology groups bHsm(M) in the (smooth) sin- gular theory. If c = P oe2map1 ( k,M)aoe oe 2 Csmk(M, R) is a smooth singular 81 4 Measure Homology is Singular Homology chain and f 2 Cbksm(M) is a singular cochain, their Kronecker product is given by hf, ci= f(c) = X aoe f(oe) 2 R. oe2map1 ( k,M) If we think of c as a linear combination of atomic measures, this looks like an integration of f over c. Hence our "dual" in measure homology will consist of (bounded) functions which can be integrated over measure chains: Definition (4.18). Let M be a smooth manifold without boundary and let k 2 N. We write fi bCk(M) := f :map 1 ( k, M) ! R fif is Borel measurable and bounded and ffi :bCk(M)! bCk+1(M) k+1 f 7 ! oe 7! ( 1) f(@(oe)) . Here f(@(oe)) is an abbreviation for P k+1j=0( 1)j f(oe ffi @j). The next lemma shows that this map ffi :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 let f 2 bCk(M). Then ffi(f) 2 bCk+1(M) and ffi ffi ffi(f) = 0. Proof.According to Remark (3.24), map 1( k+1, M) ! map 1 ( k, M) oe7 ! oe ffi @j is continuous for each j 2 f0, : :,:k + 1g. Hence ffi(f) is Borel measurable. As f is bounded, ffi(f) is bounded as well. Since @(@(oe)) = 0 holds for all oe 2 map 1 ( k+2, M), it follows that __ ffi ffi ffi(f) = 0. |__| Definition (4.20). Let M be a smooth manifold without boundary and let k 2 N. The k-th bounded measure cohomology group of M is given by bHk(M) := Hk Cb (M) , ffi . We write k k1 for the seminorm on bHk(M) which is induced by the supre- mum norm on bCk(M). 82 4.3 . . . is isometric Definition (4.21). Let M be a smooth manifold without boundary and let k 2 N. The Kronecker product of ~ 2 Ck(M) and f 2 bCk(M) is defined as Z hf, ~i:= f d~. If ~ is a measure cycle and f is a cocycle, we write ff Z [f], [~] := hf, ~i= f d~. It is easy to see that this is well-defined and - as desired - yields a general* *i- 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 , : ibCk(M) Ck(M) ! R is well-defined and bilinear. 2. The Kronecker product h , :Hibk(M) Hk(M) ! R on (co)homology is well-defined and bilinear. 3. The Kronecker product defined above is compatible with the Kronecker produ* *ct on bounded smooth cohomology in the following sense: for all f 2 bCk(M) and all c 2 Csmk(M, R), hB(f), ci= hf, iM (c)i, where B(f) denotes the linear extension of f :map 1 ( k, M) ! R to the vector space Csmk(M, R). Passage to (co)homology yields for all ' 2 bHk(M) and all ff 2 Hsmk(M, R) k ff ff H (B)('), ff = ', Hk(iM )(ff) . 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) are bounded 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) , then the transformation formula yields Z Z f ffi @jd~ = f d~(oe7!oeffi@j) map1( `,M) map1 ( ` 1,M) 83 4 Measure Homology is Singular Homology R R for all j 2 f0, : :,:`g, and hence ffi(f) d~ = f d@(~). 3. Since both Kronecker products are bilinear, it suffices to consider the case where c consists of a single smooth simplex oe 2 map 1 ( k, M). Then the left hand side - by definition - evaluates to f(oe). Since iM (c) = iM (oe) = ffioei* *s the atomic measure on map 1 ( k, M) concentrated in oe, we obtain for the right hand side Z hf, iM (c)i= f dffioe= 1 f(oe). The corresponding equality in (co)homology follows because B :bC(M) ! bCsm(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 smooth manifold without boundary, k 2 N, and ff 2 Hk(M) . 1. If kffkmh= 0, then for all ' 2 bHk(M) h', ffi= 0. 2. If kffkmh> 0, then n 1 fi o kffkmh ~ sup ______fifi' 2 bHk(M) , h', ffi=.1 k'k 1 Proof.Let ' 2 bCk(M). Assume that ~ 2 Ck(M) is a measure cycle represent- ing ff and f 2 bCk(M) is a cocycle representing '. If h', ffi= 1, then 1 = jh', ffij = jhf, ~ij fiZ fi Z Z = fifif d~fifi~ jfj d~+ + jfj d~ Z Z ~ kfk1 1 d~+ + kfk1 1 d~ = kfk1 k~k . Taking the infimum over all representatives results in 1 ~ k'k 1 kffkmh. In particular, if there exists such a ', then 1 k ffkmh~ ______> 0. k'k 1 __ Now the lemma is an easy consequence of this inequality. |__| 84 4.3 . . . is isometric A posteriori we will be able conclude - in view of Theorem (4.14), and Lemma (4.25) - that in the first part of the lemma "if and only if" is also true 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 bounded functions in map (Mek+1, R) which are ss1(M)-invariant. Then the key to the proof of Theorem (4.14) is a careful analysis of the diamond I (M) A D bC(M) bC (M) , B E Cbsm(M) the maps being defined as follows: o For k 2 N let sk: map ( k, M) ! map ( k, eM) be a Borel section of the map induced by the universal covering map. The existence of such a section is guaranteed by the following theorem: Theorem (4.24). Let M be a smooth manifold without boundary with uni- versal covering map p: Me ! M, and let k 2 N. 1. Then the map k k P :map , eM ! map , M oe7 ! p ffi oe is a local homeomorphism. 2. Moreover, there exists a Borel section of P. The (elementary, but rather technical) proof of this theorem is exiled to Subsection 4.3.3. 85 4 Measure Homology is Singular Homology If f 2 Ik(M) , we write k A(f): map 1 , M ! R oe7 ! f (sk(oe))(e0), : :,:(sk(oe))(ek) , using the inclusion map 1 ( k, M) ,! map ( k, M). o Similarly (see also Section 2.5), we define D(f) for f 2 Ik(M) as the linear extension to Ck(M, R) of the map k Sk(M) = map , M ! R oe7 ! f (sk(oe))(e0), : :,:(sk(oe))(ek) . o The map B is given by linear extension (cf. Lemma (4.22)). o 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 bridge between bH (M) and bHsm(M) ) with D and E, which both induce isometric isomorphisms 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 ~ kfk1 for all f 2 Ik(M) and all k 2 N. This implies fl fl 8,2Hk(I(M)) flH (A)(,)fl1 ~ k,k1 . 3. The diagram commutes, that is B ffi A = E ffi D. 4. The homomorphism H (B): bH (M) ! bHsm(M) induced by the cochain map B is surjective. Proof.1. The fact that B and E are well-defined and compatible with the respective 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 and that D is a cochain map. The same calculations show that the definition of A is independent of the choice of sk and that A is a cochain map. Let 86 4.3 . . . is isometric f 2 Ik(M) . Then A(f) is obviously bounded. Moreover, A(f) is measurable: By construction, A(f) is the composition k k sk k f map 1 , M ,! map , M ! map , eM ! Mek+1 ! R oe7! oe o 7! (o(e0), : :,:o(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 oe 2 map 1 ( k, M) we obtain B ffi A(f) (oe)= A(f) (oe) = f (sk(oe))(e0), : :,:(sk(oe))(ek) = D(f) jM (oe) = E ffi D(f) (oe). 4. Due to Corollary (2.38) and Theorem (4.7), the chain maps D and E induce isomorphisms 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 ff 2 Hsmk(M, R). Since iM : Csm (M, R) ! C (M) is norm preserving, it is immediate that kHk(iM )(ff)kmh~ kffksm1. The proof of the reverse inequality is split into two cases: fl fl # Suppose flHk(iM )(ff)flmh = 0. From Lemma (4.22) and Lemma (4.23) we obtain ff ff Hk(B)('), ff = ', Hk(iM )(ff) = 0 for all ' 2 bHk(M) . By the previous lemma, bHk(B) is surjective. Hence 8_2Hbksm(M) h_, ffi= 0, 87 4 Measure Homology is Singular Homology implying kffksm1= 0. fl fl # Let flHk(iM )(ff)flmh > 0. In this case, Lemma (4.23) and Lemma (4.22) yield fl fl n 1 fi ffo flHk(iM )(ff)fl~ sup ______fifi' 2 bHk(M) , ', H (i )(ff) = 1 mh k'k k M 1 n 1 fi ffo = sup ______fifi' 2 bHk(M) , Hk(B)('), ff =.1 k'k 1 We will compare the last set with the corresponding set of Corollary (4.8): Let _ 2 bHksm(M) such that h_, ffi= 1. Since the composition Hk(E) ffi Hk(D) is an isometric isomorphism (Corollary (2.38) and Theorem (4.7)), there ex- ists a , 2 Hk I (M) satisfying k k H (E) ffi H (D) (,) = _ and k,k1 = k_k 1. Then ' := Hk(A)(,) 2 bHk(M) possesses the following properties: o By construction, k k k k Hk(B)(') = H (B) ffi H (A) (,) = H (E) ffi H (D) (,) = _, and hence ff Hk(B)('), ff = h_, ffi= 1. o Furthermore, we get from Lemma (4.25) fl fl k'k1 = flHk(A)(,)fl1 ~ k,k1 = k_k 1. Combining these properties with the above estimate results in fl fl n 1 fi ffo flHk(iM )(ff)fl ~ sup ______fifi' 2 bHk(M) , Hk(B)('), ff = 1 mh k'k 1 n 1 fi o ~ sup ______fifi_ 2 bHksm(M), h_, ffi=.1 k_k 1 Since kffksm1~ kHk(iM )(ff)kmh> 0, we can use Lemma (4.8) to conclude __ kHk(iM )(ff)kmh~ kffksm1. |__| 88 4.3 . . . is isometric 4.3.3 Existence of a Borel section To complete the proof of Theorem (4.14), we still have to provide a proof of Theorem (4.24): Proof (of Theorem (4.24)).Since M is smooth, we may assume that Me is also a smooth manifold and p is a local diffeomorphism. Moreover, the smooth manifold M can be equipped with a Riemannian metric. Then there is a Riemannian metric on Me such that p is a local isometry (e.g., the pulled back Riemannian metric). The topologies on M and Me are induced by the metrics dM and dMe which are given by the respective Riemannian metrics. Then p is also a local isometry with respect to dM and dMe. 1. Let oe 2 map ( k, eM). Since p: Me ! M is a covering map, there is a small neighbourhood U of oe(e0) in eM, which means: Definition (4.26). An open subset U ae Me is called small if p(U) ae M is open and pjU :U ! p(U) is a homeomorphism. We will show that P(Ufe0g) is open and that PjUfe0g:Ufe0g ! P(Ufe0g) is a homeomorphism. fe0g The set P(Ufe0g) is open in map ( k, M). By definition, P(Ufe0g) ae p(U) . fe0g On the other hand, for each o 2 p(U) there exists a lift eo: k ! Me such that eo(e0) 2 U since k is simply connected. Thus fe0g P(Ufe0g) = p(U) . Since p(U) is open in M, this is an open subset of map ( k, M). fe0g The restriction PjUfe0g:Ufe0g ! p(U) = P Ufe0g is bijective. Since U is small, pjU is injective. Hence the uniqueness of lifts (prescribed on e0 by the property to map into U) shows injectivity ( k is connected). fe0g The restriction PjUfe0g:Ufe0g ! p(U) = P Ufe0g is a homeomorphism. Because of Lemma (3.15), the map P is continuous. It therefore remains to prove that the restriction PjUfe0gis open: Let % 2 Ufe0g. Since k is compact and p ffi % is continuous, the set K := p ffi %( k) is compact. 89 4 Measure Homology is Singular Homology Then the "distance" of the sheets of p over K is uniformly bounded from below in the following sense: Lemma (4.27). There is a constant C 2 R>0 satisfying the following property: the distance of all points y1, y2 2 p 1(K) such that p(y1) = p(y2) and y1 6= y2 is bounded from below via dMe(y1, y2) ~ C. Proof (of Lemma (4.27)).We consider the function D :K ! R x 7 ! inf dMe(ex, y), y2p 1(x)nfexg where exis some p-lift of x. The function D is independent of this choice since the fundamental group ss1(M) acts isometrically on Me and transitively on each 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 a trivialisation 'W W ss1(M) ,= p 1(W) prW 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 exin p 1(W) (the fibre ss1(M) car- ries the discrete topology) and hence in eM. So there is an rx 2 R>0 such that e the open ball BMe2rx(ex) is contained in 'W (W f1g) and p BM2rx(ex) = BM2rx(* *x). Let z be a point in the open neighbourhood Vx := BMrx(x) of x. We have e BMrx(x) ae W because pj'W (W f1g)is isometric and BM2rx(x) = p BM2rx(ex) . Since 'W is injective and BMe2rx(ex) ae 'W (W f1g), we conclude Me 'W (z, h) 62 B2rx(ex) for all h 2 ss1(M) n f1g. In particular, dMe 'W (z, h), 'W (z, 1)~ dMe 'W (z, h), ex dMe ex, 'W (z, 1) ~ 2 rx dM (x, z) ~ 2 rx rx = rx 90 4.3 . . . is isometric for all h 2 ss1(M) n f1g. This yields D(z) ~ rx for all z 2 Vx. Since K is compact, it can be covered by finitely many such sets Vx1, : :,:Vxn. Then C := min rxj j2f1,:::,ng __ gives the desired estimate. |__| We will show below that for all small enough " 2 R>0 the image of n fi o Ve":= o 2 map ( k, eM) fifisupd e(o(x), %(x)) < " M x2 k under PjUfe0gis the set n fi o V" := o 2 map ( k, M) fifisupdM (o(x), p ffi %(x)) < " . x2 k As k is compact, the topologies on map ( k, eM) and map ( k, M) are the same as the respective topologies of uniform convergence (Remark (3.13)). This proves that PjUfe0gis open. The idea is to apply the above lemma in the following way: since the sheets of the covering are far away from each other and functions in V" are very close, their lifts must lie in the "same" sheet. Using the fact that p is a loc* *al isometry, 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 ffi 2 R>0 such that for each y 2 p 1(K) the restriction Me Me pjBMe : Bffi(y) ! p Bffi(y) ffi(y) to the open ball of radius ffi around y (with respect to dMe) is an isometric d* *iffeomor- e phism and p BMffi(y) = BMffi(p(y)). Proof (of Lemma (4.28)).For each x 2 K we choose an element ex2 p 1(x). e Then there are balls BMe"x(ex) such that pjBMe : BMe"(ex) ! p BM"(ex) is an "x(ex) x x isometric diffeomorphism and p BMe"x(ex) = BM"x(x). Since ss1(M) acts by isometries on eM(see Corollary (5.10)), we conclude that Me Me g B"x(ex) = B"x(g ex) 91 4 Measure Homology is Singular Homology and that Me Me pjg BMe :g B" (ex) ! p B" (ex) "x(ex) x x is an isometric diffeomorphism for all g 2 ss1(M). As K is compact, we can find an n 2 N and x1, : :,:xn 2 K such that [ K ae BM"x /2(xj). j j2f1,:::,ng Then "xj ffi := min ___ j2f1,:::,ng2 is the desired constant: namely, let y 2 p 1(K) and x := p(y). Then there is a j 2 f1, : :,:ng such that x 2 BM"x (xj). Since ss1(M) acts transitively on j/2 e each fibre, there is a g 2 ss1(M) satisfying y 2 g BM"x(xej). Because pj Me j B"xj(* *g exj) is an isometry, "xj dMe(y, g exj) = dM (x, xj) ~ ___. 2 In particular, BMeffi(y) ae BMe"x(g exj). But then j Me Me pjBMe : Bffi(y) ! p Bffi(y) ffi(y) is also an isometric diffeomorphism. Moreover, Me BMffi(x) ae BM"x(xj) = p B" (xej) , j xj e __ since x 2 BM"x (xj). Therefore p BM (y) = BM (p(y)). |__| j/2 ffi ffi How small does " have to be? We take " 2 0, min(C/4, ffi/3) so small that eV"ae Ufe0g, where C is the constant of Lemma (4.27) and ffi is constructed in Lemma (4.28). Since " < ffi, we conclude that P(Ve") ae V": Let o 2 Ve"and x 2 k. Since %(x) 2 p 1(K) and " < ffi, it follows that dM p ffi o(x), p ffi %(x) = dMe o(x), %(x) < ". Hence P(o) = p ffi o 2 V". 92 4.3 . . . is isometric Conversely, let o 2 V". Since k is simply connected, there is a lift eoof o. Moreover, we may choose eo(e0) in such a way that dMe eo(e0), %(e0) < " because %(e0) 2 p 1(K) and dM o(e0), p ffi %(e0) < " < ffi. In order to show that eo2 eV"we have to check that the set k fi D := x 2 fidMe eo(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: o The set D is open. Let x 2 D. Since eoand % are continuous, there is an open neighbourhood W of x such that 8y2W dMe eo(y), eo(x) < " and dMe %(y), %(x) < ". Hence dMe eo(y), %(y)~ dMe eo(y), eo(x) + dMe eo(x), %(x) + dMe %(x), %(y) < " + " + " ~ ffi for all y 2 W. Now the construction of ffi yields dMe eo(y), %(y) = dM o(y), p ffi %(y) < ". Thus W ae D, implying that D is open. o The set D is closed. Suppose x 2 k n D, i.e., dMe(eo(x), %(x)) ~ ". The condition " < ffi assures that p is isometric on the "-ball in Me e around %(x) and that p BM"(%(x)) = BM"(p ffi %(x)). In particular, the ball BMe"(%(x)) contains a point y such that p(y) = o(x) because o 2 V". Now Lemma (4.27) yields dMe eo(x), %(x)~ dMe eo(x), y dMe y, %(x) ~ C ". By continuity, the set W ae k of all z satisfying dMe(eo(z), eo(x)) < " and dMe(%(z), %(x)) < " is an open neighbourhood of x. Therefore, we obtain for all z 2 W dMe eo(z), %(z)~ dMe eo(x), %(x) dMe eo(z), eo(x) dMe %(x), %(z) ~ C " " " ~ ". Hence W ae k n D, which shows that D is closed. 93 4 Measure Homology is Singular Homology This proves the first part of Theorem (4.24). 2. As for the second part note that we can cover map ( k, eM) with countably many open sets (Vn)n2N on which P is a homeomorphism (e.g., one could take a countable covering of Me by small sets U and consider sets of the form Ufe0g). In particular, the sets (P(Vn))n2N are open in map ( k, M). Via W0 := P(V0) and [ 8n2N Wn+1 := P(Vn+1) n Wj j2f0,:::,ng we get a countable family (Wn)n2N of mutually disjoint Borel subsets in map ( 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 Measure Homology and Singular Homology" [Bowen ] are outlined, showing that his approach does not work. In this section, we will use the notation of his preprint. Bowen's idea is to first improve measure cycles (without changing the rep- resented class in homology) and then explicitly construct a singular chain representing the same class. Both steps are designed in such a manner that the 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 chain homotopic to the identity. However, it is easy to see that R is not even a chain map in general. Furthermore, the inductive construction of simplices for the second step is not possible: e.g., let oe be the 2-simplex wrapping around S2 and mapping the boundary @ 2 to the point x0 2 S2. Let o be the 2-simplex which maps everything to x0 (in particular, oej@ 2 = oj@ 2). Then oe ' o relative to the vertices of 2, but not oe ' o relative to the boundary @ 2. This example also shows that, in general, [j(c)] 6= [~]: Let ~ := j(oe o). Th* *is is a cycle and ~ generates H2 S2, R ,=R. However, oe 2 H(o). Therefore, j(c) = ~ H(oe) j(o) = (1 1) j(o) = 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 to introduce an integration process for measure chains to get information about the measure homology classes represented by certain cycles. Definition (4.29). Let M be a smooth manifold without boundary. Let k 2 N, oe 2 map 1( k, M) and ! be a smooth k-form on M. Then there is an extension (f, V ) of oe and we define Z Z ! := f !, oe k 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 boundary of dimension n, we get for all oe 2 map 1 ( n+1, M) via the generalised Stokes Theorem [Lee2 , Theorem 16.10] Z Z volM = d volM @oe oe (where volM is the volume form on M). Since d volM is an (n + 1)-form of the n-dimensional manifold M, it must be zero. Hence Z volM = 0. @oe 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 Z [c]7 ! volM c on homology. 95 4 Measure Homology is Singular Homology The following lemma reveals the reason for using the C1-topology on the mapping space map 1 ( k, M) instead of the coarser compact open topology. The map described in the Lemma would not be continuous when working with the compact open topology (since the pulled back form considerably depends 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 k map 1 , M ! R Z oe7 ! ! oe is continuous. Proof.This is an immediate consequence of the definition of the C1-topology and the compatibility of integration with certain limits [Ratcliffe, Lemma_3_ in 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 k : map 1 , M ! R Z oe7 ! ! oe 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 is bounded. The mapping space map ( k, M) is Hausdorff since M is Hausdorff. Thus map 1 ( k, M) is Hausdorff by Lemma (3.23). In particular, the compact set D is closed and hence Borel. This implies that __ := OD is a bounded measurable function (where OD stands for the characteristic function of the Borel set D). Moreover, map 1 ( k, M) n D must be a ~-null set 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 map 1 ( k, M) = P [ N is a Hahn decomposition for ~. Therefore the characteristic function OD is ~+ -integrable and ~ -integrable. Since is almost everywhere (with respect to both ~+ and ~ ) dominated by the integrable function OD max x2D (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 Z ! := ! d~(oe). ~ map1 ( k,M)oe 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 Z ! = ! iM (c) 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 singular simplices oe 2 Csmk(M, R). By definition of iM , Z Z Z ! = ! d iM (oe) (o) iM (oe) map1 ( k,M)o Z = ! 1, oe __ which proves the lemma. |__| The above compatibility statement and Lemma (1.10) can be combined to yield the following description of the top measure homology group: 97 4 Measure Homology is Singular Homology Theorem (4.35). Let M be an oriented closed connected Riemannian manifold of dimension n, and let ~ 2 Cn (M) be a measure cycle. Then the measure homology class [~] can be expressed as follows: R ~ volM [~] = ________ [M]mh 2 Hn (M) . vol(M) Proof.The transformation formula shows [Ratcliffe, Lemma 4 in x11.5] Z Z Z 8 2Cn+1(M) volM = d volM = 0 = 0. @ Thus integration of the volume form induces a homomorphism Hn (M) ! R Z [ ]7 ! volM . Denoting the canonical inclusion Csmn(M, R) ,! Cn (M) by iM , we obtain the commutative diagram (compare Lemma (4.34)) R volM Hsmn(M, R) R. Hn(iM ) R volM Hn (M) 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 volM = vol(M) c holds, whenever c 2 Csmn(M, R) is a representative of [M]sm . Hence Z Z volM = volM = vol(M) ~ c for each measure cycle ~ representing the measure homology fundamental __ class [M]mh . |__| 98 The Proportionality 5 Principle _________________________________________________________________________ The proportionality principle of simplicial volume reveals a fascinating connec* *tion between the simplicial volume and the Riemannian volume: simplicial volume and Riemannian volume are proportional if we restrict our attention to manifolds wh* *ich share the same universal Riemannian covering space. Similar proportionality pri* *n- ciples also occur in the setting of L2-invariants (Theorem (1.41)) and in the s* *etting of characteristic numbers of complex manifolds [Hirzebruch, Satz 3]. Both Thurston and Gromov sketched (dual) proofs for the proportionality princip* *le of simplicial volume [Thurston, page 6.9], [Gromov2 , Section 2.3]. Thurston's * *idea is to use measure homology to compute the simplicial volume and to take advantage of the larger chain complex for the so-called smearing construction. In this ch* *apter, a detailed version of Thurston's proof will be given, based on the results of t* *he previous chapters. Gromov's proof uses arguments from bounded cohomology and also depends on some averaging operation via Haar measures. Unfortunately his exposition is not very explicit about certain measurability issues. In Section 5.1, the proportionality principle is discussed and the strategy of * *proof is presented. Sections 5.2 and 5.3 provide the necessary tools for Thurston's smea* *ring technique (i.e., the study of isometry groups and appropriate measures on them)* *. In Section 5.4, the smearing of smooth singular chains is developed. The actual pr* *oof of the proportionality principle is given in Section 5.5. The chapter ends with* * some applications 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 and N be oriented closed connected Riemannian manifolds with isometrically isomor- phic universal Riemannian coverings. Then _k_Mk___ kNk = _______. vol(M) 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 Riemannian covering 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 preserving isometry 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 in later sections. Setup (5.2). Let M and N be two oriented closed connected Riemannian manifolds with common universal Riemannian covering U. The covering maps 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 chain map 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)__ kMk ~ kNk vol(M) 100 5.1 Statement of the proportionality principle would easily follow. Swapping the r^oles of M and N would give the reverse inequality and hence prove the proportionality principle. Integration (for smooth simplices) shows that the naive approach oe 7 ! pM ffi eoe (eoebeing some pN -lift of oe) would yield the factor vol(N)/ vol(M) if it were a 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 ss1(N) is not finite in general, there is no direct way to do this. Using measure homology chains instead of singular chains, we can take advantage of the Haar measure on the locally compact group Isom+1(U) of orientation preserving isometries on U, which contains ss1(N). However, to satisfy the finiteness condition for measure chains, the compact quotient ss1(M) n Isom+1(U) is more appro- priate. To guarantee the norm condition, the constructed measure on the quotient ss1(M) n Isom+1(U) is scaled to a probability measure. The resulting chain map smear N,M :Csm (N, R) ! C (M) indeed does not increase the seminorm and induces the desired factor on the level 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 based on 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 cited which never appeared in the suggested form). A more detailed discussion of smearing is given in Ratcliffe's book [Ratcliffe, x11.5]. The proportionality constant in the proportionality principle is only known in very few special cases such as hyperbolic manifolds (Theorem (1.23)) or flat manifolds (Corollary (5.26)). 101 5 The Proportionality Principle 5.2 Isometry groups _________________________________________________________________________ As indicated in Section 5.1, the principal underlying tool for smearing is the measure on a quotient of the group of orientation preserving isometries on the universal covering manifold induced by the Haar measure on this isometry 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. A map 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 a metric isometry [Helgason , Theorem I 11.1]. Definition (5.4). Let U be a Riemannian manifold (without boundary, but not necessarily compact). o The group of (metric) isometries on U, written as Isomco(U), is the set of 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) . o 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 given by composition. The topology on Isom+1(U) is the subspace topology induced by the C1-topology on map 1 (U, U). From the topologist's point of view it is of course much more comfortable to work with the group Isomco(U). However, we are forced to use the finer C1-topology on the space of smooth singular simplices to make the inte- gration process in Lemma (4.31) continuous. Therefore - as can be seen in Lemma (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 topology and the C1-topology on the isometry groups coincide. But there does not seem 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 applying these 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 given topology. 2. The set Isomco(U) is closed in map (U, U) . 3. The group Isomco(U) is Hausdorff, locally compact, and satisfies the second countability 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. In particular, x 6= y. Since d: U U ! R~0 is continuous, there are open neighbourhoods 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" Wfyg is an open neighbourhood of f in map (U, U) satisfying fxg fyg V " W " Isomco(U) = ;, by construction of V and W. Parts 3. and 4. follow since Riemannian manifolds are second countable proper metric spaces [Sauer , Theorem 2.35]. The cited proof is based on the 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 canonical way: 103 5 The Proportionality Principle 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 ffv= (pv, qv): [ 1, 1 ] ! TU, ffw= (pw , qw:)[ 1, 1 ] ! TU representing v and w respectively, i.e., pv(0) = p = pw (0), qv(0) = q = qw (0) and v = ff0v(0), w = ff0w(0). Then the inner product of v and w is given by ff hv, wi(p,q):= T(p,q)ss(v), T(p,q)ss(w) p + hD0qv, D0qw ip where ss :TU ! U is the bundle projection, D0qv denotes the covariant derivative of the vector field qv along pv (with respect to the Riemannian connection, of course), and analogously for D0qw . Lemma (5.7). Let U be an oriented Riemannian manifold and f 2 Isom+1 (U). Then Tf 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 Tf preserves 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 ss ffi Tf = f ffi ss, we conclude T(f(p),Tpf(q))ss T(p,q)Tf(v) = Tpf T(p,q)ss(v) . Since f is an isometry of the Riemannian manifold U, this yields ff T(f(p),Tpf(q))ss T(p,q)Tf(v) , T(f(p),Tpf(q))ss T(p,q)Tf(w) f(p) ff = Tpf T(p,q)ss(v) , Tpf T(p,q)ss(w) f(p) = hT(p,q)ss(v), T(p,q)ss(w)ip. 104 5.2 Isometry groups Considering the second summand in the definition of hv, wi(p,q), we observe that (by naturality of the Riemannian connection under isometries [Lee1 , Proposition (5.6)]) D0(Tf ffi qv) = Tf(D0qv) which implies (since f is an isometry) ff ff D0(Tf ffi qv), D0(Tf ffi qw )=f(p)Tf(D0qv), Tf(D0qw ) f(p) = hD0qv, D0qw ip. Thus the Riemannian metric on TU is preserved by the map Tf, i.e., Tf is__ an 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 topo* *l- 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 map 1 (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) ae Isomco(TU). Therefore + T :Isom+1 (U) ! T Isom1 (U) ae 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 the subset 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 the subset T Isom+1(U) . Therefore Isom+1(U) is also Hausdorff. By the same argument, Isom+1(U) is second countable. 105 5 The Proportionality Principle 3. Using the fact that Isomco(TU) is Hausdorff and second countable, we can check the closedness of T Isom+1(U) via sequences: Let (gn)n2N ae 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 ae Isom+1(U). The convergence of the sequence (Tfn)n2N to g implies that (fn)n2N converges to __ g:= ssU ffi g ffi iU in the compact open topology (where iU :U ! TU is the inclusion of the zero-section and ssU :TU ! U denotes the bundle projection). Since __ iU and ssU are smooth, it follows that g is smooth. Theorem (5.5) implies __ thus that g is an isometry. __ To convince ourselves that Tg = g, we use (special) charts of TU to reduce the 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 V1 __ __ is compact) and let (V2, '2) be a chart around g(x) 2 U such that g(V1) ae V2. Since each of the gn = Tfn preserves fibres, the same must be true for their limit g. Then 1 1 g ssU (V1) ae ssU (V2). 1 By definition of the manifold structure on TU, the pairs ssU (V1), T'1 and __ ssU1 (V2), T'2 are charts of TU around each point in ssU1 (x) and ssU1 (g(x)) respectively. Since V1 is relatively compact and V2 is open, fn(V1) ae V2 holds for all large enough n 2 N. So, without loss of generality, we may assume that this is true for all n 2 N. This implies 1 1 Tfn ssU (V1) ae ssU (V2) for all n 2 N since Tfn is fibre-preserving. Thus we may define G := T'2 ffi g ffi (T'1) 1, Fn := T'2 ffi Tfn ffi (T'1) 1. 106 5.2 Isometry groups In particular, we get commutative diagrams of the form (with d := dim (U)) gnj=Tfnj 1 ssU1 (V1) ssU (V2) T'1 T'2 Fn d '1(V1) Rd '2(V2) R . On the other hand, the differential Tfn is characterised by the fact that the above diagram commutes when the map on the bottom is substituted by '1(V1) Rd ! '2(V2) Rd 1 1 (y, z)7 ! ('2 ffi fn ffi '1 )(y), J('2 ffi fn ffi '1 )y(z) (here J stands for the Jacobian matrix). Hence Fn must be this (smooth) map (the vertical arrows are diffeomorphisms). Moreover, (Fn)n2N converges to 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 the end of the proof of the theorem): Lemma (5.9). Let d 2 N, let V1, V2 ae Rd be two open sets, and (fn)n2N ae map 1 (V1, V2). Assume that the maps (Fn)n2N ae map 1 V1 Rd, V2 Rd, defined by Fn :V1 Rd ! V2 Rd (y, z)7 ! fn(y), (Jfn)y(z) converge in the compact open topology to a function G :V1 Rd ! V2 Rd. 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 ffi G ffi i1, G2 := p2 ffi G. Since the image of the zero-section is closed in TU and preserved by all the gn = Tfn, the same must hold for g. Then a straightforward computation shows __ 1 '2 ffi gffi ('1) = p1 ffi G ffi i1 =: G1. 107 5 The Proportionality Principle Therefore the lemma shows __ 8(y,z)2'1(V1) Rd(Jg )y(z) = p2 ffi G(y, z) =: G2(y, z), yielding the commutative diagram gj 1 ssU1 (V1) ssU (V2) T'1 T'2 (G1,JG1) '1(V1) Rd '2(V2) Rd. __ On the other hand, we already know that gis smooth. So the above diagram proves __ gjss 1 = Tg j 1 . U (V1) ssU (V1) __ Moreover, the map g is orientation preserving: A local diffeomorphism be- tween oriented smooth manifolds is orientation preserving if and only if its Jacobian matrix with respect to any oriented smooth charts has positive de- terminant [Lee2 , Exercise 13.5]. Using the same argument as above, we get __ that 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 fn __ preserves 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 Isom1 (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, it is clear that G ae 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 compact set K ae 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 ffi f(x) 2 K. We consider the set fi + KG := T(gf ffi f) fif 2 Isom1 (U) . According to Lemma (5.7), we know that all the elements of KG have Lip- schitz constant 1. Moreover, fi + KG (0x) := T(gf ffi f)(0x) fif 2 Isom1 (U) fi 1 ae 0y fiy 2 ss (K) = i(K), where ss :TU ! U denotes the bundle projection, 0y (for y 2 U) is the zero element of the tangent space TyU and i :U ! TU is the zero section. Thus KG (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 map 1 (TU, TU) , it follows that KG " T Isom+1(U) is relatively compact in T Isom+1(U) . Hence TjI1som+ (KG ) = fgf ffi f j f 2 Isom+1(U)g 1(U) must be relatively compact in Isom+1(U). Denoting the canonical projection Isom+1(U) ! G n Isom+1(U) by ssG , we get by construction that the restriction ssG jTj 1 (K ):Tj 1 + (KG ) ! G n Isom+1(U) Isom+1(U)G Isom1(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 coincides with 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 for their limit G. If K ae V1 is compact, the product K Sd 1 is a compact subset of V1 Rd. Hence the Fn converge uniformly on K Sd 1 to G for n ! 1. This implies that Jfn ! G2 109 5 The Proportionality Principle uniformly on K for n ! 1 (with respect to the operator norm). A standard theorem from calculus [Lee2 , Theorem A.70] now shows that G1 is 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 boundary and U its Riemannian universal covering. Then ss1(M) can be considered a sub- group of Isom+1(U) and the quotient ss1(M) n Isom+1(U) is compact. Proof.Covering theory shows that ss1(M) acts on U from the right by (mu- tually distinct) homeomorphisms. Since the covering map ssM : U ! M is an orientation preserving local isometry and ff U U ssM ssM M id M is commutative for each ff 2 ss1(M), it follows that ss1(M) acts on U by orientation preserving isometries. Moreover, multiplication in ss1(M) cor- responds to composition in Isom+1(U) [Massey , Corollary V 7.5]. Hence ss1(M) is a subgroup of Isom+1(U). The quotient ss1(M) n Isom+1(U) is compact: By covering theory, ss1(M) acts properly on U [Massey , page 136] and ss1(M) n U ,=M is compact [Massey , Lemma 8.1 in Chapter V]. Thus ss1(M) n Isom+1(U) is compact by the last __ part of Theorem (5.8). |__| Using the results of Theorem (5.8), we will now be able to conclude that the compact open and the C1-topology on the isometry groups coincide (but we will not need this fact in the sequel). Unfortunately, I was not able to figure out a proof of this fact which avoids the nasty computations in the proof of Theorem (5.8). Definition (5.11). For a Riemannian manifold U, we write Isom1 (U)for the group of isometries endowed with the subspace topology induced by the C1-topology on map 1 (U, U). Theorem (5.12). Let U be a Riemannian manifold. Then the topologies on the groups 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 that Isom1 (U) is a topological group which is Hausdorff, locally compact, and second 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 that of a standard Borel space, which provides a convenient setting for measure theory. As locally compact second countable Hausdorff groups, both Isomco (U) and Isom1 (U) are standard Borel groups [Kechris, Theorem 5.3]. Since the C1-topology is finer than the compact open topology, the identity map id: Isom1 (U) ! Isomco(U) is a bijective Borel map. Then id already is a Borel isomorphism [Kechris, Theorem 14.12]. Hence Isom1 (U) and Isomco (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. Haar measures 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 oe-algebra of the quotient ss1(M) n Isom+1(U). This measure will be used in the averaging process mentioned in the introduction. Definition (5.13). Let G be a locally compact Hausdorff topological group, and let A be its Borel oe-algebra. o A (positive) measure ~ on A is called regular if fi ~(A) = sup ~(K) fiK compact, K ae A , fi ~(A) = inf ~(U) fiU open, A ae U holds for all A 2 A, and if ~(K) is finite for all compact sets K ae G. 111 5 The Proportionality Principle o 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. o The group G is called unimodular if each left invariant Haar measure on G also is right invariant. It is well-known that such a Haar measure exists for each locally compact Hausdorff 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 the Borel oe-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 action is a Borel set F ae X such that ~(X n G F) = 0, and 8g2Gnf1g ~(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 oe-algebra of ss1(M) n Isom+1(U) satisfying hM ss1(M) n Isom+1(U) = 1. Proof.Using properties of standard Borel spaces, it can be shown that each discrete 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 ss1(M) ae Isom+1(U) is discrete and acts cocom- pactly on Isom+1(U) (Corollary (5.10)), we conclude that ss1(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 ss1(M) n Isom+1(U) [Elstrodt, Korollar 3.25 in Kapitel VIII]. If F is a finite measure fundamental domain of ss1(M) and ssM : Isom+1(U) ! ss1(M) n Isom+1(U) is the projection, then one can show that 1 h0(A) = h ssM (A) " F holds for all Borel subsets A ae ss1(M) n Isom+1(U): Indeed, for each g 2 ss1(M) the translate F g is also a measure fundamen- tal domain of finite measure because h is right invariant. Left invariance of h yields that h ssM1 ( ) " F is independent of the chosen fundamental domain F. Then it is easy to see that h ssM1 ( ) " F is a right invariant me* *a- sure on the quotient ss1(M) n Isom+1(U). Since ss1(M) has finite covolume in Isom+1(U), it follows that h0 ss1(M) n Isom+1(U) is finite. Why is h0 ss1(M) n Isom+1(U) 6= 0? Since ss1(M) is countable and F is a measure fundamental domain, we get (where the num- ber of non-zero summands might be infinite) + h Isom1 (U) = X h(g F) g2ss1(M) = X h(F). g2ss1(M) In addition, h Isom+1 (U) 6= 0 since Haar measures are by definition non- trivial. Thus + 0 6= h(F) = h0 ss1(M) n Isom1 (U) . So h0 hM := _______________________+ h0 ss1(M) n Isom1 (U) __ is a measure with the desired properties. |__| 113 5 The Proportionality Principle 5.4 Smearing _________________________________________________________________________ Using the methods of the preceding sections, we are now able to introduce the main construction - the "smearing" of smooth singular chains. In this section we will always presuppose the notation from Setup (5.2). As a first step, we define for each smooth simplex oe on U a measure smear M(oe) which is supported on all pM ffi o, where o is an (orientation preserving) iso- metric translate of oe and which is uniformly distributed - the simplex oe is "smeared" over the set pM ffi Isom+1(U) oe . 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 oe 2 map 1 ( k, U). The map k Foe:ss1(M) n Isom+1(U) ! map 1 , M ss1(M) f7 ! pM ffi f ffi oe 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 continuous map. Proof.For all ff 2 ss1(M) we get pM ffi ( ff) = pM since ss1(M) acts by deck transformations, implying that Foeis independent of the chosen representative. By Lemma (3.15) and Remark (3.24) the map k Isom+1(U) ! map 1 , M f7 ! pM ffi f ffi oe is continuous. Thus the universal property of the quotient topology yields __ the continuity of Foe. |__| 114 5.4 Smearing Definition (5.17). For each k 2 N and each oe 2 map 1 ( k, U), we define the smeared chain of oe by smearM (oe) := hFoeM, where hM is the measure from Theorem (5.15). We extend smear M linearly to the whole chain group Csmk(U, R). By Lemma (5.16), the map Foeis, in particular, Borel measurable. Hence smear M (oe) is a well-defined measure on map 1 ( k, M). Lemma (5.18). 1. For each k 2 N and each smooth chain c 2 Csmk(U, R) we have smear M(c) 2 Ck(M) . 2. The map smear M :Csm (U, R) ! C (M) is a chain map. Proof.1. Of course, we need only show this assertion for smooth singular simplices oe 2 map 1 ( k, U). According to Lemma (3.4), hM is a signed mea- sure of finite total variation because hM ~ 0 and + hM ss1(M) n Isom1 (U) = 1. Furthermore, ss1(M) n Isom+1(U) is a compact determination set of hM . Now Lemma (3.9) yields smearM (oe) = hFoeM2 Ck(M) since Foeis, due to Lemma (5.16), continuous. 2. Let k 2 N, j 2 f0, : :,:k + 1g, and let oe 2 map 1( k+1, U) be a smooth singular (k + 1)-simplex. By linearity, it is sufficient to show smear M (oe ffi @j) = @j smear M (oe) where oe ffi @jstands for the j-th face of oe. By construction, Foeffi@j smear M (oe ffi=@j)hM (ss1(M)) f7!pM ffifffioeffi@j = hM F (o7!offi@ ) = hMoe j F = @j hMoe __ = @j smear M (oe) . |__| To pull back the smearing operation to smooth singular simplices on N via pN -lifts, we need the right invariance of the measure hM to guarantee well- definedness: 115 5 The Proportionality Principle Lemma (5.19). Let k 2 N and oe 2 map 1 ( k, N). 1. Then there is a pN -lift of oe lying in map 1 ( k, U). In fact, each conti* *nuous pN -lift of oe is smooth. 2. If oe1, oe2 2 map 1 ( k, U) are two pN -lifts of oe, then Foe1 Foe2 hM = hM . Proof.1. Since k is simply connected, there is a continuous pN -lift eoe: k ! U of oe. Noting that the covering projection pN :U ! N is a local diffeomor- phism (even a local isometry), we deduce from the commutative diagram U eoe pN k oe N that eoealso must be smooth (smoothness is a local property). 2. Since k is connected, two continuous pN -lifts of oe only differ by some element in ss1(N). As already mentioned in Corollary (5.10), the fundamen- tal group ss1(N) acts by orientation preserving isometries on the universal cover U. Hence there is a g 2 Isom+1(U) such that oe1 = g ffi oe2. Writing rg: ss1(M) n Isom+1(U) ! ss1(M) n Isom+1(U) for right multiplica- tion by g, we get Foe1= Foe2ffi rg. Thus, Foe1 Foe2ffirg rg Foe hM = hM = hM .2 Finally, the right invariance of hM enters the scene: Since hM is right invar* *i- ant, we have rg hM = hM , __ which completes the proof. |__| Hence we can pull back smear M to smooth singular chains on N: 116 5.4 Smearing Definition (5.20). For each k 2 N and each smooth singular simplex oe 2 map 1 ( k, N) we define its smeared chain by smear N,M (oe) := smear M(eoe) 2 Ck(M) , where eoe2 map 1 ( k, U) is some pN -lift of oe. Again, the notation smear N,M will also be used for its linear extension Csm (N, R) ! C (M) . Lemma (5.21). The smearing map smear N,M :Csm (N, R) ! C (M) is a chain map. Proof.Let k 2 N, j 2 f0, : :,:k + 1g, and let oe 2 map 1 ( k+1, N). If eoeis a pN -lift, then eoeffi @j is obviously a pN -lift of oe ffi @j. Hence we conclud* *e from Lemma (5.18) smear N,M (oe ffi=@j)smearM(eoeffi @j) = @j smear M (eoe) = @j smear N,M(oe) . __ Thus smear N,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 smear N,M(oe) for all smooth singular simplices oe 2 map 1 k, N . This implies k ksmear N,M(oe)k= smear N,M(oe) map 1( , U) + = hM ss1(M) n Isom1 (U) = 1. Now the general case is an immediate consequence of the triangle inequality_ for the total variation (compare Lemma (3.7)). |__| The integration operation introduced in Section 4.4 makes it possible to un- derstand the effect smear N,M has on homology: 117 5 The Proportionality Principle Theorem (5.23). Smearing induces multiplication by vol(N)/ vol(M) on the top homology group. More precisely: for all cycles z 2 Csmn(N, R) with [z] = [N]sm in Hsmn(N, R), vol(N) smear N,M(z) = ________ [M]mh 2 Hn (M) . vol(M) The proof basically relies on the following calculation: Lemma (5.24). For all oe 2 map 1 ( k, N) the following integrals are equal: Z Z volM = volN. smearN,M(oe) oe Proof (of Lemma (5.24)).Let eoebe some pN -lift of oe. By definition, the integ* *ral can be computed as follows: Z Z Z volM = volM d smear N,M (oe) (o) smearN,M(oe) map1 ( k,M)o Z Z = volM d hFeoeM(o). map1 ( k,M)o Using the transformation formula for measures, we get Z Z Z volM = volM dhM (ss1(M) f) smearN,M(oe) Zss1(M)nIsom+1(U)Feoe(ss1(M)Zf) = volM dhM (ss1(M) f) Zss1(M)nIsom+1(U)pMZffifffieoe = f pM volM dhM (ss1(M) f). ss1(M)nIsom+1(U)eoe By definition, pM and pN are orientation preserving local isometries. Since orientation preserving local isometries between Riemannian manifolds pull 118 5.4 Smearing back the volume form to the volume form, we conclude Z Z Z volM = volU dhM (ss1(M) f) smearN,M(oe) Zss1(M)nIsom+1(U)eoeZ = pN volN dhM Zss1(M)nIsom+1(U)eoe Z = dhM pN volN ss1(M)nIsom+1(U)Z eoe = 1 volN Z pNffieoe __ = volN. |__| oe Proof (of Theorem (5.23)).Lemma (5.24) and Lemma (1.10) imply Z Z volM = volN = vol(N). smearN,M(z) z Application of Theorem (4.35) therefore results in vol(N) __ smear N,M(z) = ________ [M]mh 2 Hn (M) . |__| vol(M) Remark (5.25). It is possible to extend the smearing chain map smear N,M to all measure chains: Ck(N) ! Ck(M) ` Z ' ~ 7 ! A 7! hFeoeM(A) d~(oe) . map1 ( k,N) 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 for the proportionality principle of the "simplicial volume" defined via measure homology. 119 5 The Proportionality Principle 5.5 Proof of the proportionality principle _________________________________________________________________________ Combining the techniques from Section 5.4 and the isometric isomorphism of Section 4.3, we are now in state to do the final step in the proof of the proportionality 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 smear N,M induces the factor vol(N)/ vol(M) on top homology, i.e., vol(N) smearN,M (z) = ________ [M]mh . vol(M) 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) ~ ________ ksmear N,M(z)k vol(N) vol(M) sm ~ ________ kzk1 . vol(N) Now taking the infimum over all (smooth) representatives of [N]sm gives the estimate vol(M) sm vol(M) kMk ~ ________ k[N]smk1 = ________ kNk vol(N) vol(N) because k[N]sm ksm1= k Nk by Corollary (4.4). Swapping the r^oles of the manifolds M and N shows the reverse inequality vol(N) kNk ~ ________ kMk vol(M) __ 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 give now a proof of this fact based on the proportionality principle instead of applying 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., all lengths are stretched by the factor c). Then M and Mc are homeomorphic manifolds (in particular, kMk = kMck ) and vol(Mc) = cn vol(M) where n is the dimension of M. Moreover, Mc is also flat. Therefore both universal covers Me and fMcare isometric to Rn [Lee1 , Theorem 11.12]. Thus the proportionality principle yields _kMk____ kMck kMk = ________ = ___________. vol(M) vol(Mc) 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 from the proportionality principle that the Riemannian volume in some special cases is homotopy invariant: Corollary (5.27). 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In Theorem 3.4 of this paper a way is described how to derive the corresponding results and proofs for the smooth theory. Like Zastrow we refer to those adapted statements of his paper by * *num- bers such as, e.g., 4.5diffinstead of 4.5. 126 Table of Notation _________________________________________________________________________ Symbols________________________ kMk simplicial volume 5 # connected sum kMk C 21 ffl one point space kMk Q 21 ` wedge kMk Z 21 h , i Kronecker product 29, k k1 seminorm 26, 27, 74, 82 R 30, 75, 83 k k1 supremum norm 26, 74 ~! integration of the [x] (co)homology class form ! over the represented by x R measure chain ~ 97 [M] fundamental class 5 oe! integration of the form [M] R fundamental class with ! over the smooth real coefficients 5 singular simplex oe 95 [M] R fundamental class with k 56 R-coefficients 21 k 1k `1-norm 5 [M] mh measure homology k 1k seminorm 5 fundamental class 80 k Rk1 "seminorm" 21 [M] sm fundamental class in k skm1 seminorm on smooth smooth singular singular homology 72 homology 73 k mkh seminorm for measure chains 62 k k total variation of a A________________________________ signed measure 51 A 85 127 Table of Notation B________________________________ ffi coboundary operator B 83, 85 3, 74, 82 B(G, V ) set of bounded @j j-th face of a singular functions from G to V simplex 2 36 @j j-th face of a standard BMr(x) open ball in M of simplex 2 radius r around x 90 @j homorphism induced BMer(x) open ball in eMof by the j-th face map 61 radius r around x 90 dM metric on M 89 dMe metric on eM 89 C________________________________ C complex numbers E________________________________ C (X) = C (X, Z) 3 E 85 C (X, R) singular chains of X with coefficients in R 2 F________________________________k C (f, R) homomorphism F (X) set of bounded induced by f 4 functionsek+1 C (M) measure chains 61 in map (X , R) 44 C (M, N) relative measure chain Fg oriented closed group 62 connected surface of C (X) = C (X, Z) 3 genus g 10 C (X, R) singular cochains of X Foe 114 with coefficients in R 3 H_______________________________ C (f, R) homomorphism H (X) = H (X, Z) 4 induced by f 4 H (f) = H (f, Z) 4 bC(X) set of bounded H (X, R) singular homology cochains in X 26 of X with coefficients colim colimit/direct limit in R 3 bCk(M) bounded Borel H (f, R) homomorphism measurable functions induced by f 4 map1 ( k, M) ! R 82 H (M) measure homology bCsm(M) bounded smooth of M 62 cochains of M 74 H (f) homomorphism Csm (M, R) set of smooth singular induced by f 63 simplices in M 72 H (M, N) relative measure cX comparison map 30 homology group 62 Hsm (M, R) smooth singular D________________________________ homology of M 72 D 85 H (X) = H (X)Z 4 @ boundary operator 3, H (f) = H (f, Z) 4 61, 72 H (X, R) singular cohomology deg degree of a map 8 of X with coefficients k standard k-simplex 2 in R 3 128 Table of Notation H (f, R) homomorphism M_______________________________ induced by f 4 map (X, Y) set of continuous maps Hb (X) bounded cohomology from X to Y 54 of X 26, 38 map co1(M, N)set of all smooth maps Hb (f) homomorphism from M to N with the induced by f 27, 38 compact open Hb (M) bounded measure topology 54 cohomology of M 82 map co1( k, M) 57 Hbsm(M) bounded smooth map 1 (M, N) set of all smooth maps cohomology 74 from M to N with the hM 112 C1-topology 54 map 1 (M, k) 56 map 1 ( k, M) 56 I_________________________________ minvol (M) minimal volume 13 I (X) set of ss1(X)-invariant ~f pushforward of ~ with functions in F (X) 44 respect to f 53 iM inclusion Csm (M, R)! C (M) 76 N_______________________________ im image N nonnegative integers Isom1 (U) group of isometries, endowed with the P________________________________ C1-topology 110 ss1 fundamental group Isomco(U) isometry group with pM 100 the compact open pN 100 topology 102 Isom+1(U) group of orientation Q_______________________________ preserving isometries, Q rational numbers endowed with the C1-topology 102 R________________________________ R real numbers J_________________________________ jM inclusion S________________________________ Csm (M, R),! Sk(X) set of singular C (M, R) k-simplices in X 2 72 s smoothing operator 73 secg(M) sectional curvature K________________________________ of M with respect to ker kernel the Riemannian metric g 13 smear M first step of smearing L________________________________ 115 k 56 smear N,M smearing 117 129 Table of Notation T________________________________ T k = k Rk 57 Tf differential of f U________________________________ u 45 UK 54 V________________________________ V" 91 eV" 91 V G G-fixed points in V 37 vn maximal volume of an ideal n-simplex in hyperbolic space 12 vol(M) volume of M volM volume form of M Z________________________________ Z integers 130 Index _________________________________________________________________________ A________________________________ smooth 74 amenable 39, 39, 43 bounded G-module 35 aspherical 17 bounded measure cohomology 82 bounded smooth cochain 74 B________________________________ bounded smooth cohomology 74 Borel function 53 duality principle of 76 Borel oe-algebra 52 Kronecker product for 75 boundary 3 boundary operator 3, 61 C________________________________ bounded cochain 26 C1-topology 54, 58 bounded cohomology 26 canonical seminorm 38 characteristic classes and 42 chain dimension axiom 28 measure 61 duality principle of 29 singular 2 excision 42 smeared 115, 117 first group 33 characteristic classes 42 fundamental group and 40 coboundary 3 homotopy invariance of 28 coboundary operator 3, 74, 82 Kronecker product for 30 cochain naturality of 27 bounded 26 of groups 35, 38 bounded smooth 74 simplicial volume and 32 singular 3 131 Index cocycle 3 in measure homology 80 cohomology in singular homology 5 bounded in smooth singular homology 73 see bounded cohomology fundamental theorem of homological bounded measure 82 algebra 37 singular 3 compact open topology 54, 55 G________________________________ comparison map 30 G-module C1-topology 55 bounded 35 connecting homomorphism for mea- relatively injective 36 sure homology 64 G-morphism 35 contracting homotopy 36 strongly injective 35 cycle 3 G-resolution 36, 37, 38, 44, 45 relatively injective 36 D________________________________ standard 36 decomposition strong 36 Hahn 51 group action ek+1 Jordan 51 on map (X , R) 44 degree 8 on B(G,nV ) 36 determination set 50 on B(G , R) 36 differential 57 dimension axiom H_______________________________ bounded cohomology 28 Haar measure 112 measure homology 63 Hahnhdecompositionomology 51 duality 29 measure 62 duality principle singular 3 of bounded cohomology 29 smooth singular 72 of bounded smooth cohomology homotopy invariance 76 of bounded cohomology 28 of measure homology 84 of measure homology 64 of simplicial volume 9 E________________________________ Hopfian 19 excision bounded cohomology and 42 I_________________________________ for measure homology 64 induced measure 53 extension 56 integrable 52 integral 52 F________________________________ integration face 2 and volume 6 flat manifold 121 measure homology and 98 functorial seminorm 24 of measure chains 95, 97 functoriality 8, 23 of smooth singular simplices 6, fundamental class 95, 95 132 Index isometric isomorphism 31, 39, 40, 42, Mayer-Vietoris sequence 65 45, 72, 75, 80, 81, 85 naturality of 63 isometry 102 relative 62 isometry group 102, 102, 103, 105, simplicial volume and 81 110, 110 singular homology and 81 measure on 112 minimal volume 13 Mostow rigidity theorem v J_________________________________ multiplicativity 8, 9, 19 Jordan decomposition 51 ~-negative 50 ~-positive 50 K________________________________ Kronecker product 29 N_______________________________ for bounded cohomology 30 naturality for bounded smooth cohomology of bounded cohomology 27 75 of measure homology 63 for measure homology 83, 83 negative variation 51 normal 65 L________________________________ null set 50 L2-invariants 17 left invariant 112 P________________________________ long exact pair sequence for measure positive variation 51 homology 64 proportionality principle 120 applications 121 2 M_______________________________ of L -invariants 19 mapping theorem 42 of simplicial volume 13, 100 Mayer-Vietoris sequence for measure toy version of 12 homology 65 pushforward of a signed measure 53 measure chain 61 integration of 95, 97 R________________________________ relative 62 regular 111 measure fundamental domain 112 relative measure chain group 62 measure homology 60, 62 relative measure homology 62 boundary operator of 61 relatively injective colimits and 66 G-module 36 connecting homomorphism 64 G-resolution 36 dimension axiom 63 duality principle of 84 S________________________________ excision for 64 seminorm 5, 21, 23, 26, 27 fundamental class 80 bounded measure cohomology homotopy invariance of 64 82 integration and 98 canonical 38 Kronecker product 83, 83 measure homology 62 long exact pair sequence 64 smooth bounded cohomology 74 133 Index smooth singular homology 72 T________________________________ signed measure 50 topology simplex C1-topology 54, 55 singular 2 compact open 54, 55 smooth singular 72 on Isom1(U) 110 standard 2 on Isom+1(U) 102 simplicial volume 5 on Isomco(U) 102 amenability and 43 on map (X, Y) 54 bounded cohomology and 32 on map co1(M, N) 54 classifying maps and 43 on map 1(M, N) 54 fudamental group and 43 on map co1( k, M) 57 generalisations 21 on map 1( k, M) 58 homotopy invariance of 9 rocks! L2-invariants and 17 see topology measure homology and 81 total variation 51, 51, 52 of Sn 9 of connected sums 44 U________________________________ of flat manifolds 14, 121 unimodular 112 of hyperbolic manifolds 12 universal Riemannian covering 100 of products 34 of S1 7 V________________________________ of surfaces 10, 13 variation 51 proportionality principle 13, 100 negative 51 realisability of 15 positive 51 S1-actions and 15 total 51, 51, 52 singular chain 2 volume singular cochain 3 minimal 13 singular cohomology 3 simplicial singular homology 3 see simplicial volume measure homology and 81 singular simplex 2 integration of 6, 95, 95 small 89 smearing 101, 114, 115, 117, 117, 118 smooth 56 smooth singular homology 72 smooth singular simplex 72 integration of 6, 95, 95 standard resolution 36 standard simplex 2 strong 36 strongly injective 35 supremum norm 26, 74 134