Dipartimento di Matematica
Corso di Dottorato in Matematica
Ph.D. Thesis
On Gromov’s theory of multicomplexes:
the original approach to
bounded cohomology and simplicial volume
Advisor:
Prof. Roberto Frigerio
Candidate:
Marco Moraschini
XXXI Ph.D. cycle
Pisa, 2018
“Karma police arrest this man he talks in maths he buzzes like a fridge he is like a detuned radio”. (Radiohead - Karma police)
ACKNOWLEDGEMENTS
First of all, I wish to thank my advisor Roberto Frigerio, since this thesis would not be possible without his constant, friendly and expert guidance. I am grateful to him not only for introducing me into a such a nice topic, but also for many enlightening discussions. Only with his constant support and his advices I was able to conclude this long journey. Beyond mathematics, I would like to thank him for many friendly moments spent together. Overall, I still have to learn lots of stuff from him, for instance learning to play chess well.
Moreover, I wish to thank the topology group in Pisa. Especially, I am indebted with Riccardo Benedetti, Roberto Frigerio, Bruno Martelli and Carlo Petronio for their very nice courses on advanced topology and for organizing great seminars. A special thank goes to Bruno Martelli for the wonderful conferences he organized during the past years. I also wish to thank Mario Salvetti (and again Roberto, Bruno and Riccardo) for the teaching experiences shared together.
I warmly thank Clara L¨oh and Roman Sauer for serving as referees. I thank Clara for her useful comment on triangulable manifolds and Roman for pointing to me an interesting and challenging application of this thesis.
I am also grateful to Davide Ferrario for introducing me to topology from the very first courses to the advanced ones during my undergraduate studies in Milan. His nice courses easily convinced me to study topology and thank to his uncountable tips and suggestions I had the opportunity to study here in Pisa.
I wish to thank the topology team of M´alaga which gave a fundamental contri-butions to my background in algebraic topology during my undergraduate studies. A special thank goes to Aniceto Murillo for friendly introducing me to the most algebraic and abstract part of topology. I also wish to thank David, Luis, Kiko and Pepe not only for their constant help and suggestions during these years, but also to be enough imaginative to understand my written Spanish.
During my Ph.D. I had the opportunity of sharing mathematics with a lot of people that I wish to thank here. The first thank goes to the other organizers of the
Edoardo Fossati and Nicoletta Tardini. In particular, I thank Carlo and Nicoletta for getting me involved in the organization. Moreover, a special mention is due to Edoardo for a lot of inspiring discussions about topology, which began during our first undergraduate exam Geometria I. Finally, I am indebted with the director of the Ph.D. program Giovanni Alberti for founding this event.
I wish to thank Federico Franceschini for some useful discussions about the topics of this thesis.
Some special mentions are due to my oldest officemates. First I wish to thank Stefano Riolo for all his precise and enlightening answers about hyperbolic geometry. Remember that Stephen has the solution to all our problems! I also wish to thank Andrea Vaccaro not only for choosing appropriate soundtracks for our office, but also for letting me always win at dards! Thanks also to Sabine not only for having dinner at italian time, but also for our favourite ice creams. Last but now least, I thank to Carlo Collari not only for the countless discussions about topology but also for introducing me to Arkham Horror.
I am also grateful to the ufficio degli analisti, which always answered to all my analytic questions with (too much) patience. A special thank goes to Giacomo Del Nin for all his tips and inspiring discussions.
I also thank Nikita Simonov for many philosophical discussion on mathematics. Special thanks are also due to my conference friends Marco Barberis, Caterina Campagnolo, Luigi Caputi, Filippo Cerocchi, Elia Fioravanti, Nicolaus Heurer, Erika Pieroni, Giuseppe Pipoli, Leone Slavich, Davide Spriano and i bolognesi (Gianluca Faraco, Lorenzo Ruffoni and Alessio Savini).
A very special mention is due to my friends (before colleagues) here in Pisa who shared with me a lot of nice moments during these years. Special thanks are due to Marta not only for going to many concerts with me, but also for a lot of curtigghio. I also wish to warmly thank Alessandra not only for being our rappresentante, Alessio not only for the organization of many GEPPETTO, Annar`ı not only for her broccoli, Chiara and Federico not only for worms, Cricci not only for the PHC, Francesco not only for improving my siciliano skills, Frank not only for her tapanedda, Giulio not only for the passo-svelting, Leonardo not only for his violin, Marco not only for the “D”, Matteo not only for the beach volley, Oscar not only for LATEX tips, Sabbbino
not only for his cinghiale al salm`ı, Valerio not only for hosting my cassoeula! Thanks to Daniele for introducing me to storpionimi, I promise you that one day we will be able to cook a proper tortilla!
I wish to express all my gratitude to my parents for encouraging and support-ing me dursupport-ing these years. Thanks also to my brother and Amanda not only for swimming with me in the almost frozen lakes of Lapland.
Finally, the warmest thank goes to Laura for constantly encouraging me with her lovely presence. Moving in with her in our casupola in Pisa made these years an unforgettable experience.
ABSTRACT
The simplicial volume is a homotopy invariant of manifolds introduced by Gro-mov in his pioneering paper Volume and bounded cohomology. In order to study the main properties of simplicial volume, Gromov himself initiated the dual theory of bounded cohomology, which then developed into a very active and independent research field. Gromov’s theory of bounded cohomology of topological spaces was based on the use of multicomplexes, which are simplicial structures that generalize simplicial complexes without allowing all the degeneracies appearing in simplicial sets.
The first aim of this thesis is to lay the foundation of the theory of multicom-plexes. After setting the main definitions, we construct the singular multicomplex K(X) associated to a topological space X, and we prove that the geometric real-ization of K(X) is homotopy equivalent to X for every CW complex X. Following Gromov, we introduce the notion of completeness, which, roughly speaking, trans-lates into the context of multicomplexes the Kan condition for simplicial sets. We then develop the homotopy theory of complete multicomplexes, and we show that K(X) is complete for every CW complex X.
In the second part of this thesis we apply the theory of multicomplexes to the study of the bounded cohomology of topological spaces. Our constructions and arguments culminate in the complete proofs of Gromov’s Mapping Theorem (which implies in particular that the bounded cohomology of a space only depends on its fundamental group) and of Gromov’s Vanishing Theorem, which ensures the vanishing of the simplicial volume of closed manifolds admitting an amenable cover of small multiplicity.
The third and last part of the thesis is devoted to the study of locally finite chains on non-compact spaces, hence to the simplicial volume of open manifolds. We expand some ideas of Gromov to provide complete proofs of a criterion for the vanishing and a criterion for the finiteness of the simplicial volume of open manifolds. As a by-product of these results, we prove a criterion for the `1-invisibility of closed
CONTENTS
Introduction i
Chapter 0. Background 1
0.1. Amenability 1
0.2. Bounded cohomology 4
0.3. Simplicial volume of compact manifolds 9
0.4. Duality Principle and applications 14
0.5. Simplicial volume of open manifolds 15
0.6. `1-homology and its applications to simplicial volume 17
Part I. The general theory of multicomplexes 19
Chapter 1. Multicomplexes 21
1.1. Basic definitions 21
1.2. The geometric realization 24
1.3. Multicomplexes, simplicial complexes, ∆-complexes, and simplicial sets 26
1.4. Simplicial (bounded) (co)homology 30
1.5. Group actions on multicomplexes 36
Chapter 2. The singular multicomplex 39
2.1. The weak homotopy type of the singular multicomplex 41
2.2. (Lack of) functoriality 49
Chapter 3. The homotopy theory of complete multicomplexes 51
3.1. Complete multicomplexes 52
3.2. Special spheres 54
3.3. Simplicial approximation of continuous maps 59
3.4. Minimal multicomplexes 62
volume 73
Chapter 4. Bounded cohomology of multicomplexes 75
4.1. Bounded cohomology of complete multicomplexes 76
4.2. Amenable groups of simplicial automorphisms 82
4.3. Bounded cohomology is determinated by the fundamental group 89
4.4. Multicomplexes and relative bounded cohomology 95
ADDENDUM 98
4.A. An alternative proof of Theorem 4.1.2 98
Chapter 5. The Mapping Theorem 103
5.1. The group Π(X, X0) 104
5.2. The action of Π(X, X) on A(X) 105
5.3. Proof of Gromov’s Mapping Theorem 112
Chapter 6. The Vanishing Theorem 115
6.1. Vanishing Theorems 115
6.2. Amenable subgroups of Π(X, X) and their action on A(X) 117
6.3. Proof of Vanishing Theorem II 120
6.4. Proof of Vanishing Theorem I 122
Part III. The simplicial volume of open manifolds 125
Chapter 7. The Finiteness and the Vanishing Theorems 127
7.1. The Vanishing and the Finiteness Theorems 128
7.2. An application to `1-invisibility 129
Chapter 8. Diffusion of chains 131
8.1. Diffusion operators 132
8.2. Locally finite actions and diffusion 136
8.3. A toy example 140
Chapter 9. Admissible submulticomplexes of K(X) 145
9.1. (Strongly) admissible simplices and admissible maps 145
9.2. Admissible multicomplexes 148
9.3. Group actions on the admissible multicomplex 150
9.4. Amenable subgroups of AutAD(AD0L(X)) 158
Chapter 10. The proofs of the Vanishing and the Finiteness Theorems 167
10.1. An important locally finite action 168
10.3. Proof of the Vanishing Theorem 7.1.3 174
10.4. Proof of the Finiteness Theorem 7.1.4 175
Chapter 11. Some results on the simplicial volume of open manifolds 177
11.1. PL manifolds 177
11.2. Locally coamenable subcomplexes 178
11.3. The simplicial volume of the product of three open manifolds 182
Bibliography 185
INTRODUCTION
Simplicial volume. The simplicial volume is an invariant of manifolds intro-duced by Gromov in his pioneering paper Volume and bounded cohomology [Gro82]. If X is a topological space and Sn(X) denotes the set of singular n-simplices with
values in X, then the space Cn(X) of singular n-chains with real coefficients is
endowed with the `1-norm defined by X σ∈Sn(X) aσσ = X σ∈Sn(X) |aσ| .
This norm descends to a seminorm on the homology H∗(X) with real coefficients,
and if M is a closed oriented n-dimensional manifold, then the simplicial volume kM k of M is the `1-seminorm of the real fundamental class [M ] ∈ H
n(M ).
While being defined only in terms of singular chains, the simplicial volume is deeply related to many invariants of geometric nature. As Gromov stated in the in-troduction of [Gro82], “The main purpose of this paper is to provide new estimates from below for the minimal volume in terms of the simplicial volume” (the mini-mal volume of a Riemannian manifold M is the infimum of the volume of complete Riemannian metrics on M subject to the condition that the absolute value of all sectional curvatures is not bigger than 1). The simplicial volume vanishes for man-ifolds admitting a Riemannian metric with non-negative Ricci curvature, and it is positive for negatively curved manifolds [Gro82]. For closed hyperbolic manifolds, a fundamental result by Gromov and Thurston shows that kM k = Vol(M )/vn, where
vn is a constant only depending on the dimension n of M [Thu79, Gro82]. This
result (which provides one of the few exact computations of the simplicial volume) describes the hyperbolic volume explicitly in terms of a topological invariant, and plays a fundamental role in a celebrated proof of Mostow Rigidity Theorem due to Gromov and Thurston.
Applications and open questions. There are by now many results which relate the simplicial volume to invariants and phenomena of differential geometric nature. Gromov proved in [Gro82] that the non-vanishing of the simplicial volume implies the non-vanishing of the minimal volume and of the minimal entropy, an invariant which, roughly speaking, measures the rate of growth of balls in the uni-versal covering. Since then, the simplicial volume has been extensively exploited in the study of volumes of balls and of systolic inequalities in Riemannian manifolds (see e.g. [Gut11]).
Moreover, Gromov’s Proportionality Principle [Gro82] ensures that the ratio kM k/ Vol(M ) between the simplicial volume and the classical volume of a closed Riemannian manifold only depends on the isometry type of the universal covering of M (see also [Thu79, L¨oh06, BK08a, Fri11, Fra16, Str17]). Lafont and Schmidt proved in [LS06] that the proportionality constant between kM k and Vol(M ) is positive for every locally symmetric space of non-compact type (see also [BK07] for a case not covered by Lafont and Schmidt’s argument). Thus locally symmetric spaces of non-compact type have positive simplicial volume, and this answers a question posed by Gromov [Gro82].
Other long-standing questions on the relationship between curvature and sim-plicial volume for Riemannian manifolds are still unsolved. For example, Gro-mov conjectured in [Gro82] that the simplicial volume of a non-positively curved manifold with negative Ricci curvature should be positive. We refer the reader to [CWa, CWb] for some recent progress on this topic. The simplicial volume has also been studied in relation with the Chern Conjecture, which predicts that the Euler characteristic of a closed manifold admitting an affine structure should vanish [BG11, BCL]. The simplicial volume is a key ingredient also in the proof of a collapsing theorem which plays a fundamental role in the last step of Perelman’s proof of Thurston’s Geometrization Conjecture [BBB+10].
Also the study of the geometry of smooth maps (e.g. via the analysis of the complexity of their critical sets and of their fibers) has profited from the use of the simplicial volume [Gro09, GG12]. Recently, the simplicial volume has also been applied to count trajectories of vector fields [Kat16, AK16, Alp16].
Of course, the simplicial volume has found applications also to problems which are more topological in nature. The elementary remark that the degree of any map f : M → N between closed manifolds of the same dimension is bounded above by the ratio kM k/kN k laid the foundation for many results on domination be-tween manifolds (an orientable manifold M dominates the orientable manifold N if there exists a map f : M → N of non-vanishing degree). We refer the reader e.g. to [Der10, BRW14] for some results on domination between 3-manifolds that were obtained via the study of the simplicial volume.
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Also when dealing with the topological features of simplicial volume, it turns out that some long-standing questions are still unsolved. For example, Gromov asked in [Gro93, p. 232] whether the `2-Betti numbers of a closed aspherical manifold M with kM k = 0 should vanish (see also [Gro09, 3.1. (e) on p. 769]). If answered in the affirmative, Gromov’s question would imply, for example, that if M is an orientable aspherical closed manifold admitting a self-map of degree d 6∈ {1−, 0, 1}, then the Euler characteristic of M vanishes. In order to approach his question, Gromov himself introduced a variation of the classical simplicial volume, called the integral foliated simplicial volume [Gro99]. For some results on this topic we refer e.g. to [Sau02, Sch05, FFM12, LP16, FLPS16].
Other (even more elementary) open questions on the simplicial volume are re-lated to products and fiber bundles. If M, F are closed orientable manifolds of di-mension m, n respectively, then the simplicial volume of M × F satisfies the bounds kM k · kF k ≤ kM × F k ≤ cn,mkM k · kF k, where cn,m is a constant only depend-ing on n, m (see e.g. [Gro82]). Of course, products are just a special case of fiber bundles, so one may wonder whether the simplicial volume of the total space E of a fiber bundle with base M and fiber F could be estimated in terms of the simplicial volumes of M and F . In the case when F is a surface it was proved in [HK01] that the inequality kM k · kF k ≤ kEk still holds. This estimate was then improved in [Buc09]. It it still an open question whether the inequality kEk ≥ kM k · kF k holds for every fiber bundle E with fiber F and base M , without any restriction on the dimensions of F and M .
Finally, thank to [Sam99, Theorem 1.1], a proportionality theorem between simplicial volume and minimal entropy, we are able to extend the family of mani-folds for which the value of their simplicial volume is known. Indeed, that theorem applies to all closed manifolds whose fundamental group admits a subexponential representation into the fundamental group of some closed, negatively curved, locally symmetric manifold (see [Sam00, Section 3] for some precise computations).
Bounded cohomology. Computing the simplicial volume has proved to be a very challenging task. Besides hyperbolic manifolds, the only other closed manifolds for which the exact value of the simplicial volume is known are 4-dimensional man-ifolds locally isometric to the product H2× H2 of two hyperbolic planes [BK08b]
(for example, the product of two hyperbolic surfaces). Starting from these examples, more values for the simplicial volume can be obtained by taking connected sums or amalgamated sums along submanifolds with an amenable fundamental group.
In order to study the simplicial volume, Gromov himself developed in [Gro82] the dual thery of bounded cohomology. If ϕ ∈ Cn(X) is a singular cochain with real
coefficients, then one can define the `∞-norm of ϕ by setting kϕk∞= sup{|ϕ(σ)| , σ ∈ Sn(X)} ∈ [0, +∞] .
A cochain is bounded if its `∞-norm is finite. It is easily seen that bounded cochains define a subcomplex Cb∗(X) of C∗(X), whose cohomology is the bounded cohomology Hb∗(X) of X. The inclusion Cb∗(X) ,→ C∗(X) induces the comparison map
c∗: Hb∗(X) → H∗(X) .
For every n ∈ N, the normed space Cbn(X) coincides with the topological dual of
C∗(X) (endowed with the `1-norm). Using this, it is not difficult to show that the
vanishing of Hbn(X) implies the vanishing of the `1-seminorm on Cn(X) (hence, of the
simplicial volume, if X is a closed n-dimensional manifold). This and other duality results have been exhaustively exploited in the study of the simplicial volume. For example, the computation of the simplicial volume of manifolds locally isometric to H2× H2 takes place in the context of bounded cohomology, and it is based on the study of bounded cocycles rather than of cycles. As a quite surprising consequence, there is no description of any fundamental cycle for the product of two hyperbolic surfaces whose `1-seminorm approximates the value of the simplicial volume.
One of the most peculiar features of the bounded cohomology of a space X is that it only depends on the fundamental group of X. In particular, the bounded cohomology of any simply connected space vanishes. In order to prove this funda-mental result (and other related results), Gromov developed in [Gro82] the theory of multicomplexes. While being based on very neat geometric ideas, Gromov’s the-ory of multicomplexes involves a certain amount of technicalities and raises some difficulties, which have never been completely overcome.
Multicomplexes have then been exploited in several papers on the simplicial volume (see e.g. [Kue15, KK15, Str]). However, all these papers give for granted several fundamental results from [Gro82] (like Theorems 1 and 2 below), whose proofs in [Gro82] are omitted or just sketched. The difficulties encountered in working with multicomplexes encouraged the interested mathematicians to develop alternative approaches to the bounded cohomology of spaces (hence, to the simplicial volume of manifolds). The study of bounded cohomology via standard tools coming from homological algebra was initiated by Brooks [Bro81] and developed by Ivanov in his foundational paper [Iva87] (and, years later and in a much wider context, by Burger and Monod [BM99, Mon01, BM02]; see also [B¨uh11]).
As stated by Ivanov in the introduction of [Iva], the need to provide a new proof of Gromov’s result on the vanishing of bounded cohomology for simply con-nected spaces was due to the fact that, when reading Gromov’s original paper, “I failed in my attempts to understand the proofs of the general results about
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the bounded cohomology, such as the vanishing of the bounded cohomology of simply-connected spaces”. Ivanov’s proof [Iva87] was based on a modification of the Cartan–Serre killing homotopy group process [CS52] and on the Dold–Thom construction [DT58], and applies only to spaces that are homotopy equivalent to countable CW complexes. As observed by Buehler in [B¨uh11], “The proof of this result is quite difficult and not very well understood as is indicated by the strange hypothesis on X (Gromov does not make this assumption explicit, his proof is how-ever rather sketchy to say the least). The reason for this is the fact that the com-plete proof given by Ivanov uses the Dold–Thom construction which necessitates the countability assumption”. Recently, Ivanov modified his proof that applies now to any topological space. Among other results, in this thesis we will give a self-contained proof of the vanishing of the bounded cohomology of simply connected spaces, following (but modifying, sometimes in a substantial way) Gromov’s original approach.
We refer the reader e.g. to [Mon06] for a description of the wide range of applications of bounded cohomology (of spaces and of groups) to different fields in geometry and algebraic topology (see also the books [Mon01, Fri17]).
Multicomplexes. The first aim of this thesis is to lay the foundation of the theory of multicomplexes. We believe that renovating the interest towards this topic could be fruitful for at least two reasons. First of all, approaching the simplicial volume via multicomplexes (rather than from the dual point of view of bounded cohomology) means working with cycles (in a quite concrete way) rather than with cocycles (via duality). In our view, this allows a more direct understanding of the topology and the geometry of cycles, and could be of help in finding new approaches to some long-standing open questions on the simplicial volume. Secondly, while bounded cohomology is very effective in dealing with finite singular chains (hence, with the simplicial volume of compact manifolds), the use of multicomplexes is still necessary in the study of locally finite chains (hence, of the simplicial volume of open manifolds) – we refer the reader to [L¨oh07, Remark C.4] for a discussion of this issue. Similarly, we hope that multicomplexes could help to understand some peculiar features of relative bounded cohomology (hence, of the simplicial volume of manifolds with boundary) – see the discussion in Section 4.4.
Most results proved in this thesis on the simplicial volume of closed manifolds admit alternative proofs which do not make use of multicomplexes. On the con-trary, here we provide the first complete proofs of several results stated by Gromov in [Gro82] on the simplicial volume of open manifolds. Indeed, in [Gro82] one may find outlines for the proofs of various results stated here. However, when trying to fill in the details of Gromov’s arguments, we often needed to face substantial difficulties
that we were not able to overcome without diverging from Gromov’s original path. In some cases, Gromov’s proofs were so concise that it is not even clear whether our work follows his ideas or not: for example, Gromov’s proof of the Vanishing and the Finiteness Theorems for the simplicial volume of open manifolds is 4 pages long (from page 60 in [Gro82]), while we devote four chapters of this thesis to these results.
A natural question is whether one could obtain the results proved in this thesis by working with simplicial sets rather than with multicomplexes. This would give the unquestionable advantage of working with well-established and well-understood simplicial structures, for which an exhaustive literature is available. However, after trying to translate Gromov’s ideas into the less exotic context of simplicial sets, we finally could not disagree with Gromov’s deep intuition that multicomplexes should be preferred as effective tools for the study of the bounded cohomology of topological spaces. We refer the reader e.g. to Remark 4.2.7 for a brief discussion of this issue. Before stating the main results proved in this thesis, let us briefly introduce the notion of multicomplex. As stated in [Gro82], a multicomplex is “a set K divided into the union of closed affine simplices ∆i, i ∈ I, such that the intersection of any
two simplices ∆i ∩ ∆j is a (simplicial) subcomplex in ∆i as well as in ∆j”. More
formally, a multicomplex is an unordered ∆-complex in which every simplex has distinct vertices, or, equivalently, a symmetric simplicial set in which every non-degenerate simplex has distinct vertices. We refer the reader to Definition 1.1.1 for the precise definition of multicomplex, and to Section 1.3 for a thorough discussion of the relationship between multicomplexes and other well-known simplicial structures. The singular multicomplex. If X is a topological space, then the singular multicomplex K(X) associated to X is a multicomplex whose n-simplices are given by the singular n-simplices with distinct vertices in X, up to affine automorphisms of the standard simplex ∆n. This definition evokes the notion of singular complex S(X) of X, which is the simplicial set having as simplices the singular simplices with values in X. The geometric realizations |S(X)| and |K(X)| of both S(X) and K(X) come equipped with natural projections j : |S(X)| → X, S : |K(X)| → X. A fundamental result in the theory of simplicial sets ensures that j is a weak homotopy equivalence. In Section 2.1 we establish the same result for the map S:
Theorem 1. Let X be a good space. The natural projection S : |K(X)| → X
is a weak homotopy equivalence.
We refer the reader to Definition 2.1.1 for the definition of good topological space. Here we just anticipate that every CW complex is good.
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Gromov states in [Gro82] that Theorem 1 can be proved by a “standard ar-gument”, and refers the reader to the paper [Moo58], which describes some appli-cations to homotopy theory of the classical theory of simplicial sets. However, due to the lack of functoriality of the singular multicomplex (see Section 2.2) we were not able to adapt the classical ideas described in Moore’s paper from the context of simplicial sets to the context of multicomplexes. Indeed, our proof of Theorem 1 is rather inspired to Milnor’s paper [Mil57]. It is also worth mentioning that The-orem 1 is stated in [Gro82] without any assumption on the topology of X. The question whether Theorem 1 could hold for any topological space is discussed in Question 2.1.8 and in Remarks 3.2.6 and 4.1.4.
The Isometry Lemma. Weak homotopy equivalences induce isometric isomor-phisms on bounded cohomology [Iva]. Therefore, by Theorem 1 we can compute the bounded cohomology of a good space X by looking at the singular bounded cohomology of |K(X)|. Just as for simplicial sets, the simplicial cohomology of a multicomplex K (which will be denoted by the symbol H∗(K)) is isomorphic to the singular cohomology of its geometric realization. There is no hope to extend this re-sult to bounded cohomology for general multicomplexes. However, Gromov showed in [Gro82] that the inclusion of bounded simplicial cochains into bounded singular cochains induces an isometric isomorphism Hb∗(K) → Hb∗(|K|) provided that the multicomplex K is complete and large. We provide a complete proof of this fact in Section 4.1:
Theorem 2 (Isometry Lemma). Let K be a complete and large multicomplex. Then for every n ∈ N there exists a canonical isometric isomorphism
Hbn(K) → Hbn(|K|) .
Complete, minimal and aspherical multicomplexes. A multicomplex K is large if every connected component of K contains infinitely many vertices. Com-pleteness is a much subtler notion, which evokes in the context of multicomplexes the Kan condition for simplicial sets (see Definition 3.1.1 and Remark 3.1.2). Complete-ness is particularly useful in the study of homotopy groups of geometric realizations. Indeed, the combinatorial description of homotopy groups of Kan simplicial sets is a classical topic in the theory of simplicial sets, and in Chapter 3 we develop a similar theory for complete multicomplexes. In particular, following Gromov, we introduce the notion of minimal multicomplex and of aspherical multicomplex, and we show that to every large and complete multicomplex K there are associated a minimal and complete multicomplex L homotopy equivalent to K, and an aspheri-cal, minimal and complete multicomplex A having the same fundamental group of K.
A crucial intuition of Gromov is that, if K, L and A are as above, then, for every n ∈ N, the n-skeleton of the aspherical multicomplex A is the quotient of the n-skeleton of L with respect to the action of an amenable group. This result ultimately depends (in a very indirect way) on the fact that A is obtained from L by killing all higher homotopy groups, and higher homotopy groups are abelian, hence amenable. The very same fact is exploited by Ivanov in his modification of the Cartan–Serre killing homotopy group process, which gives rise to a tower of weakly principal bundles with amenable (but infinite-dimensional!) structure groups. Here we develop Gromov’s intuition into self-contained and complete proofs. To this aim, we need to introduce some modifications not only to Gromov’s arguments, but even to some (fundamental) definitions given in [Gro82]. For example, our definition of the group of simplicial automorphisms Γ such that L/Γ ∼= A is different from Gromov’s. We refer the reader to Remark 4.2.2 for a detailed discussion of this issue (see also Remark 9.4.8).
The combinatorial description of the homotopy groups of a complete multicom-plex is carried out in Theorems 3.2.5 and 3.4.5, and it plays a fundamental role in many of our proofs. For example, it allows us to prove that the singular multicom-plex K(X) associated to a good topological space is complete (see Theorem 3.2.3), and to establish useful criteria to recognize completeness, minimality and asphericity of a given multicomplex (see Proposition 3.5.2). It is also essential in the proof that suitable quotients of a distinguished subgroup of the group Γ mentioned above are amenable (see Corollary 4.2.11). Theorems 3.2.5 and 3.4.5 are basically taken for granted in [Gro82], where they are never explicitly stated (but they are extensively used).
Bounded cohomology is determined by the fundamental group. The second part of this thesis is devoted to the use of multicomplexes in the study of bounded cohomology and of the simplicial volume of closed manifolds. As just men-tioned, a key result in the theory of multicomplexes is that, for every good topological space X, the singular multicomplex K(X) is complete. Therefore, putting together Theorem 1 with Theorem 2 and the invariance of bounded cohomology with respect to weak homotopy equivalences, one obtains that Hb∗(X) is isometrically isomorphic to the bounded simplicial cohomology Hb∗(K(X)) of K(X) for every good topological space X (Gromov states the same result for any topological space in [Gro82], see Remark 4.1.4 for a discussion of this issue). Moreover, one may define a complete and minimal multicomplex L(X) and a complete, minimal and aspherical multicom-plex A(X) associated to X. Since the action of an amenable group is invisible to bounded cohomology, from all these facts we can deduce that the bounded cohomol-ogy of X is isometrically isomorphic to the simplicial bounded cohomolcohomol-ogy of the
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aspherical multicomplex A(X) (which satisfies π1(|A(X)|) ∼= π1(X)). Using this,
in Section 4.3 we prove that the bounded cohomology of X only depends on the fundamental group of X:
Theorem 3. Let f : X → Y be a continuous map between path connected spaces, and suppose that f induces an isomorphism on fundamental groups. Then the in-duced map
Hbn(f ) : Hbn(Y ) → Hbn(X) is an isometric isomorphism for every n ∈ N.
As an immediate corollary we obtain the vanishing of bounded cohomology for simply connected spaces:
Corollary 4. Let X be a simply connected topological space. Then Hbn(X) = 0
for every n ≥ 1.
The Mapping Theorem. With more work, in Chapter 5 we prove in fact a stronger statement, which is usually known as Gromov’s Mapping Theorem:
Theorem 5. Let X, Y be path connected topological spaces, let f : X → Y be a continuous map, and suppose that the map f∗: π1(X) → π1(Y ) induced by f on
fundamental groups is surjective and has an amenable kernel. Then for every n ∈ N the map
Hbn(f ) : Hbn(Y ) → Hbn(X) is an isometric isomorphism.
The proof of Theorem 5 exploits the action on A(X) of the group Π(X, X), which was first defined by Gromov in [Gro82]. Every element of the group Π(X, X) consists of a family of homotopy classes (relative to the endpoints) of paths in X, subject to some additional conditions. Every normal subgroup N of π1(X) gives rise
to a normal subgroup bN of Π(X, X), and we prove in Theorem 5.2.7 that the quotient of A(X) by the action of bN is a complete, minimal and aspherical multicomplex whose fundamental group is isomorphic to π1(X)/N (this fact is stated in [Gro82,
page 47] without proof). Building on Theorem 5.2.7, with some care one can then reduce Gromov’s Mapping Theorem to Theorem 3 above.
Amenable subsets and the Vanishing Theorem. The group Π(X, X) plays a fundamental role in many results we prove in this thesis, both when studying singular chains and bounded cohomology, and when dealing with locally finite chains on open manifolds. Before switching our attention to the latter topic, let us state a vanishing theorem whose proof exploits the action of Π(X, X) just mentioned.
Let X be a topological space and let i : U → X be the inclusion of a subset U of X. Then U is amenable in X if for every path connected component U0 of U the image of i∗ : π1(U0) → π1(X) is an amenable subgroup of π1(X). A cover
U = {Ui}i∈I of X is amenable if every element of U is amenable in X. We denote by N (U ) and by mult(U ) the nerve and the multiplicity of U , so that mult(U ) = dim N (U ) + 1 (see Section 0.2).
Theorem 6. Let X be a topological space and let U be an amenable open cover of X. Then for every n ≥ mult(U ) the comparison map
cn: Hbn(X) → Hn(X) vanishes.
In fact, under additional hypotheses on the topological space X and on the cover U one may obtain a stronger result. Recall that, if X is paracompact, then for any open cover U of X there is a continuous map f : X → |N (U )|, which is uniquely determined up to homotopy (see Section 6.1). We denote by β∗: Hb∗(|N (U )|) → Hb∗(X) the map induced by f on bounded cohomology.
Theorem 7. Let X be homeomorphic to the geometric realization of a simplicial complex and let U be an amenable open cover of X. Also suppose that, for every finite subset I0 ⊆ I, the intersection
T
i∈I0Ui is path connected (possibly empty).
Then for every n ∈ N there exists a map Θn: Hbn(X) → Hn(N (U )) such that the following diagram commutes:
Hbn(X) c n // Θn Hn(X) Hn(N (U )) ∼ = // Hn(|N (U )|) . βn OO
Theorems 6 and 7 are proved in Chapter 6. Theorem 6 is originally due to Gromov [Gro82], while Theorem 7 was proved (in a slightly different formulation) by Ivanov in [Iva87, Theorem 6.2] and in [Iva, Theorem 9.1]. Ivanov’s argument is completely different from ours, and it is based on the use of a variation of the Mayer–Vietoris double complex for singular cohomology (see Remark 6.1.3 for fur-ther details). Via duality, Theorem 6 implies the vanishing of the simplicial volume for closed manifolds admitting amenable covers of small multiplicity (see Corol-lary 6.1.4):
Corollary 8. Let X be a topological space admitting an open amenable cover of multiplicity m, and let n ≥ m. Then
xi
for every α ∈ Hn(X). In particular, if M is a closed manifold admitting an open
amenable cover U such that mult(U ) ≤ dim M , then kM k = 0 .
The simplicial volume of open manifolds. The third part of this thesis is devoted to the study of the simplicial volume of open (i.e. connected, non-compact and without boundary) manifolds. If M is an open n-dimensional manifold, then Hn(M ) = 0. Therefore, in order to define a fundamental class for M , one needs to
work with the complex C∗lf(M ) of locally finite chains on M , rather than with the
usual complex C∗(M ) of finite singular chains. As in the case of finite chains, for
every topological space X and for every n ∈ N, the space of locally finite chains Cnlf(X) may be endowed with an `1-norm k · k1, which induces an `1-seminorm k · k1
on the locally finite homology Hnlf(X) of X (see Section 0.5 for the details). If M is an n-dimensional oriented open manifold, then Hlf
n(M ) is canonically isomorphic to
R and generated by a preferred element [M ] ∈ Hnlf(M ), called the fundamental class
of M . The simplicial volume kM k of M is then defined by setting kM k = k[M ]k1 ∈ [0, +∞] .
The simplicial volume of open manifolds is still quite mysterious. If M is tame, i.e. it is the internal part of a compact manifold with boundary M , then it is known that kM k ≤ kM k, and that the equality holds provided that the fundamental group of every connected component of ∂M is amenable [L¨oh08, KK15] (we refer the reader to Section 4.4 for the definition of the simplicial volume of a manifold with boundary). However, in the case when ∂M is not amenable, no example is known for which kM k < +∞ and kM k > kM k. Neither is known any example of an open manifold M with non-amenable topological ends for which kM k /∈ {0, +∞} (while we refer the reader e.g. to [BBI13, LS09b] for the exact computation of the simplicial volume of some manifolds which compactify to manifolds with an amenable boundary). For example, a long-standing open question is the following:
Question 9. Let Σ = (S1× S1) \ {p} be a once-punctured torus. What is the exact value of kΣ × Σk? In particular, is kΣ × Σk positive or is it null?
The simplicial volume of open manifolds lacks several topological and geomet-ric properties enjoyed by the simplicial volume of closed manifolds. For example, neither a biLipschitz estimate of kM1× M2k in terms of kM1k · kM2k nor a
Pro-portionality Principle holds for the simplicial volume of open manifolds (but see e.g. [LS09b, BK14] for some results in this direction) and these facts somewhat illustrate the difficulties in understanding the topological and the geometric meaning of this invariant.
Gromov himself introduced in [Gro82] some variations of the simplicial volume for open manifolds. Among them, the most studied is probably the so called Lipschitz simplicial volume, for which one can recover, for example, both a product formula and the Proportionality Principle (see e.g. [LS09a, Fra16, Str]; see also [Str17], where the additivity of the Lipschitz simplicial volume under connected sums is proved by exploiting multicomplexes). However, in this thesis we will only deal with the classical simplicial volume of open manifolds.
The Finiteness and the Vanishing Theorems for non-compact spaces. Even if exact computations are very difficult, there exist criteria that provide condi-tions for the vanishing or the finiteness of the simplicial volume of open manifolds. Just as in the closed case, the vanishing of the simplicial volume of an open manifold may be deduced from the existence of a suitable open cover of small multiplicity. Indeed, Chapters 7, 8, 9 and 10 are devoted to the following results, that were first stated in [Gro82]. We refer the reader to Definition 7.1.2 for the notion of amenability at infinity for a sequence {Ui}i∈N of open subsets of X.
Theorem 10 (Vanishing Theorem). Let X be a connected non-compact topo-logical space, and assume that X is homeomorphic to the geometric realization of a simplicial complex. Let U = {Ui}i∈N be an amenable open cover of X such that each
Ui is relatively compact in X. Also suppose that the sequence {Ui}i∈N is amenable
at infinity. Then for every k ≥ mult(U ) and every h ∈ Hklf(X) we have khk1= 0 .
A subset W of X is large if X \ W is relatively compact in X.
Theorem 11 (Finiteness Theorem). Let X be a connected non-compact topo-logical space, and assume that X is homeomorphic to the geometric realization of a simplicial complex. Let W be a large open subset of X, and let U = {Ui}i∈N be an
open cover of W such that each Ui is relatively compact in X. Also suppose that
the sequence {Ui}i∈N is amenable at infinity (in particular, U is locally finite in X).
Then for every k ≥ mult(U ) and every h ∈ Hklf(X) we have khk1 < +∞ .
The following corollaries provide the main applications of Theorems 7.1.3 and 7.1.4 to the simplicial volume of open manifolds.
Corollary 12. Let M be an oriented open triangulable manifold of dimension m and let U = {Ui}i∈Nbe an amenable open cover of M such that each Uiis relatively
compact in M . Also suppose that the sequence {Ui}i∈N is amenable at infinity, and
that mult(U ) ≤ m. Then
xiii
Corollary 13. Let X be an oriented open triangulable manifold of dimension m. Let W be a large open subset of M , and let U = {Ui}i∈N be an open cover of W
such that each Ui is relatively compact in M . Also suppose that the sequence {Ui}i∈N
is amenable at infinity (in particular, U is locally finite in M ). Then kM k < +∞ .
As mentioned above, the proof of Theorems 10 and 11 is only 4 pages long in [Gro82]. Nevertheless, much work is needed in order to provide complete proofs of these statements. After introducing diffusion of chains, one needs to select a suitable submulticomplex AD(X) of K(X) with the following property: an infinite family of simplices of AD(X) leaves every compact subset of X provided that the set of vertices of the simplices in the family do so. Such a multicomplex fits the need to relate the local finiteness of chains in X to controlled combinatorial properties of the corresponding simplicial chains in AD(X).
A peculiarity of the proofs of Theorems 10 and 11 is that they completely avoid any reference to bounded cohomology. Theorem 10 extends Corollary 8 (which deals with the usual singular homology) to the locally finite case. However, Corollary 8 was obtained via duality from Theorem 6, which concerns the comparison map defined on bounded cohomology. On the contrary, the proofs of Theorems 10 and 11 are based on manipulations of cycles. Just as in the finite case, the amenability of certain groups of simplicial automorphisms plays a fundamental role in our arguments. However, here amenability allows us to properly exploit the diffusion of locally finite chains, rather than to obtain vanishing results in bounded cohomology.
Even if inspired by Gromov’s suggestions and by the ideas developed in the previous chapters, the arguments described in the third part of this thesis are sub-stantially new. Indeed, Gromov’s strategy to prove Theorems 10 and 11 is merely sketched in [Gro82], and it seems to underestimate several difficulties that emerge e.g. in the case when X is not assumed to be aspherical (see e.g. the long proof of Proposition 9.3.7) or when proving the amenability of some relevant groups of simplicial automorphisms (see Remark 9.4.2 and the long proof of Theorem 9.4.15, which is completely omitted in [Gro82]). For example, the simplicial automor-phisms belonging to these groups are allowed to move the vertices of AD(X). As a consequence, when showing that these groups are amenable via a comparison with suitable subgroups of Π(X, X), one needs to carefully take into account the role played by basepoints, an issue probably underrated by Gromov throughout the whole paper [Gro82] (see again Remark 4.2.2).
Amenable covers and `1-invisibility. Other results on the vanishing and/or the finiteness of the simplicial volume of open manifolds were obtained e.g. in
[L¨oh08, LS09a]. In particular, L¨oh provides in [L¨oh08] a complete criterion for the finiteness of the simplicial volume of tame open manifolds, in terms of the so-called `1-invisibility of the boundary components of the manifolds (see Definition 0.6.1 and Theorem 0.6.3). As a by-product of our results, by comparing the Finiteness Theorem 7.1.4 with L¨oh’s result, in Section 0.6 we obtain a new sufficient condition for a closed manifold to be `1-invisible:
Theorem 14. Let M be a closed triangulable n-dimensional manifold admitting an amenable cover U such that mult(U ) ≤ n. Then M is `1-invisible.
Since the simplicial volume of an `1-invisible closed manifold vanishes, Theo-rem 14 strengthens (at least in the case of triangulable manifolds) the last statement of Corollary 8.
The simplicial volume of the product of three open manifolds. Let us conclude this introduction with an interesting consequence of Theorem 10 already pointed out by Gromov. As mentioned above, it is not known whether the simplicial volume of the product of two puctured tori vanishes or not. More in general, it is not known whether the simplicial volume of the product of two open manifolds M1, M2
should be positive, provided that kM1k > 0 and kM k2 > 0. Actually, there exist
no examples of open manifolds M1, M2 for which the simplicial volume kM1× M2k
is known to be positive and finite. When considering the product of three open manifolds, we have the following striking result:
Theorem 15. Let M1, M2, M3 be tame open PL manifolds of positive dimension.
Then
kM1× M2× M3k = 0 .
Theorem 15 is stated in [Gro82, page 59] as an almost direct application of the Vanishing Theorem 12. After clarifying some statements by Gromov on coamenable subcomplexes of open manifolds, in Section 11.3 we exploit a construction inspired by [LS09a, Theorem 5.3] to provide a complete proof of Theorem 15.
CHAPTER 0
BACKGROUND
This chapter introduces the reader to amenable groups, bounded cohomology and simplicial volume. Here we just collect the fundamental definitions and results we will need in the sequel. For a much broader and detailed introduction to these topics we refer the reader e.g. to [Fri17, L¨oh17, L¨oh].
Contents
0.1. Amenability 1
0.2. Bounded cohomology 4 0.3. Simplicial volume of compact manifolds 9 0.4. Duality Principle and applications 14 0.5. Simplicial volume of open manifolds 15 0.6. `1-homology and its applications to simplicial volume 17
0.1. Amenability
The notion of amenability was first introduced by Von Neumann in [vN29] to study the Banach-Tarski paradox. Since all the groups which appear in this thesis are assumed to be discrete, we specialise to the case of amenable discrete groups (see for instance [Pie84, Pat88] for amenability of locally compact groups). We present here the definition of amenable groups in terms of invariant means, referring the reader to e.g. [L¨oh17, Section 9.2] for other approaches.
Let Γ be a group. We denote by `∞(Γ) the space of bounded real valued functions on Γ. Let us consider the natural left action of Γ on itself by left multiplication. It descends to a left Γ-action on `∞(Γ):
Γ × `∞(Γ) → `∞(Γ)
(g, f ) 7→ g · f = (h 7→ f (g−1· h)).
Definition 0.1.1. A mean m on a group Γ is a linear map m : `∞(Γ) → R
satisfying the following properties
(1) Normalisation. If 1Γ denotes the constant map with value 1 ∈ R, then
m(1Γ) = 1.
(2) Positivity. If f ∈ `∞(Γ) is a non-negative function, then m(f ) ≥ 0.
We say that a mean is left-invariant (or simply invariant ) if it satisfies the following additional condition
(3) Left-invariance. For every g ∈ Γ and f ∈ `∞(Γ), m(g · f ) = m(f ). The existence of an invariant mean characterizes amenable groups. Definition 0.1.2. A group Γ is amenable if it admits an invariant mean. Example 0.1.3. The easiest examples of amenable groups are finite groups. Indeed, given a finite group Γ it is always possible to define an invariant mean as follows: m : `∞(Γ) → R f 7→ 1 |Γ| X g∈ Γ f (g).
The first non obvious class of amenable groups has been found by Von Neumann [vN29]:
Theorem 0.1.4 ([vN29]). Every Abelian group is amenable.
Starting from Abelian groups and using inheritance properties, it is possible to enlarge the class of amenable groups. More precisely, the following inheritance properties hold (we refer the reader to e.g. [L¨oh17, Section 9.1] for the complete proofs of the following results):
Theorem 0.1.5. Let Γ and H be amenable groups. Then, (i) Every subgroup of Γ is amenable;
(ii) If the sequence
1 → H → E → Γ → 1 is exact, then E is amenable.
(iii) Every direct union of amenable groups is amenable.
Remark 0.1.6. It is easy to check that in fact also the converse of the second item of the theorem above holds: if the sequence is exact and E is amenable, then Γ and H are also amenable.
0.1. AMENABILITY 3
Example 0.1.7. The second item of Theorem 0.1.5 shows that the finite direct product of amenable groups is amenable. This implies that also the infinite direct sum of amenable groups is amenable by the last item of Theorem 0.1.5. However, an infinite direct product of amenable groups need not be amenable (see Remark 0.1.12). The previous theorem readily implies the following characterisation of amenable groups:
Corollary 0.1.8. Let Γ be a group. Then, Γ is amenable if and only if all finitely generated subgroups of Γ are amenable.
As promised before, we are now able to enlarge the class of amenable groups including some well-known families of groups. Indeed, using the extension property, one can prove the following:
Corollary 0.1.9. Let Γ be a virtually solvable group, then Γ is amenable. It is not true that we can obtain any amenable group starting from an Abelian group and then applying iteratively Theorem 0.1. Indeed, let elementary amenable groups be the smallest class of groups which contains all the finite and Abelian groups and that is closed under taking subgroups, taking quotients and direct unions. Every elementary amenable group is in fact an amenable group, but the converse it is not true (we refer the reader to [L¨oh17, Overlook 9.1.9]).
We conclude this very quick survey on amenable groups by mentioning the easiest example of non-amenable group.
Theorem 0.1.10. The free group F2 of rank 2 is not amenable.
Thanks to Theorem 0.1.5 we obtain the following sufficient condition for non-amenability:
Corollary 0.1.11. A group Γ which contains the free group F2 of rank 2 as a
subgroup is not amenable.
The previous corollary provides a huge class of non-amenable groups. However, we warn the reader that it is not true that every non-amenable group must contain F2 as a subgroup as it was first conjectured by Von Neumann in [vN29]. Indeed,
Ol’shanskii constructed in [Ol’80] a non-amenable torsion group, which of course cannot contain the free group F2 of rank 2 as subgroup.
Remark 0.1.12. We anticipated in Example 0.1.7 that infinite products of amenable groups need not be amenable. Recall that a group is residually finite if it may be embedded in the infinite finite product of finite groups (see for instance [L¨oh17, Definition 4.E.1]). Since finite groups are amenable it suffices to show that
their infinite direct product contains the free group F2 of rank 2 as subgroup.
How-ever, free groups of finite rank are residually finite and so applying Corollary 0.1.11 we conclude that the infinite product of finite groups is not amenable.
0.2. Bounded cohomology
Bounded cohomology is an algebraic tool that was first introduced by Johnson [Joh72] and Trauber to face problems about Banach algebras. However, it started to develop as an independent and active research field in 1982, thanks to Gromov, who extended this notion from groups to topological spaces [Gro82].
Let X be a topological space. Recall that (C∗(X; R), ∂∗) is the real chain complex
of X and (C∗(X; R), δ∗) is the real dual cochain complex. Henceforth, we denote the previous vector spaces simply by C∗(X) and C∗(X), respectively. We endow the
cochain complex C∗(X) with an `∞-norm as follows: if f ∈ Ck(X), then kf k∞:= sup{|f (σ)| | σ is a singular k-simplex}.
We say that a cochain f ∈ C∗(X) is bounded if kf k∞< +∞. We define the bounded
cochain complex Cb∗(X) ⊂ C∗(X) to be the subcomplex of all bounded cochains of X (it is immediate to check that the coboundary of a bounded cochain is still a bounded cochain). More precisely, we set
Cb∗(X) := {f ∈ C∗(X) | kf k∞< +∞}.
Definition 0.2.1. The bounded cohomology of X, Hb∗(X), is the homology of
the bounded cochain complex Cb∗(X). Definition 0.2.2. The inclusion
Cb∗(X) ,→ C∗(X) induces a map
c∗: Hb∗(X) → H∗(X) called comparison map.
Remark 0.2.3. It is worth noting that in general the comparison map is neither surjective nor injective (see e.g. Theorems 0.2.10 and 0.2.11).
Since the real bounded cochain complex (Cb∗(X), δ∗) is endowed with an `∞ -norm, we can induce a seminorm on the quotient as follows: given ϕ ∈ Hbk(X), we have:
kϕk∞= inf{kf k∞| [f ] = ϕ, f ∈ Zbk(X)}
0.2. BOUNDED COHOMOLOGY 5
Using the comparison map ck: Hbk(X) → Hk(X) we define kψk∞ ∈ [0, +∞] for
a singular cochain ψ ∈ Hk(X) as follows:
kψk∞= inf{kϕk∞| ϕ ∈ Hb∗(X), c ∗
(ϕ) = ψ}, where inf(∅) = +∞.
It is readily checked that any continuous map f : X → Y induces a norm non-increasing map on bounded cohomology
Hbn(f ) : Hbn(Y ) → Hbn(X).
Moreover, homotopic maps induce the same map on bounded cohomology. Using these facts, it is not difficult to prove that a homotopy equivalence induces isometric isomorphisms in every degree on bounded cohomology. In fact, a recent result by Ivanov shows that even weak homotopy equivalences (i.e. maps which induce isomorphisms on all homotopy groups) induce isometric isomorphisms on bounded cohomology:
Theorem 0.2.4 ([Iva, Corollary 6.4]). Let f : X → Y be a weak homotopy equivalence. Then the map
Hbn(f ) : Hbn(Y ) → Hbn(X) is an isometric isomorphism for every n ∈ N.
It is well known that in the case of ordinary cohomology there exists a strong connection between singular cohomology of groups and singular cohomology of a par-ticular class of spaces. Indeed, a classical result of Eilenberg and Mac Lane [EM45] states that the singular cohomology of an aspherical CW complex (see below for a precise definition) is canonically isomorphic to the singular cohomology of its fun-damental group. One of the most surprising features of bounded cohomology is not only that this result still holds, but also that it can be improved. To this end, we have to extend the definition of bounded cohomology to groups:
Definition 0.2.5. Let Γ be a group. A topological space X is called an Eilenberg-MacLane space of type K(Γ, 1) (or, simply aspherical ), if its fundamental group is isomorphic to Γ and all other homotopy groups of X are trivial.
Definition 0.2.6. We define the bounded cohomology , Hb∗(Γ), of a group Γ to be the bounded cohomology of an Eilenberg-MacLane space of type K(Γ, 1).
Remark 0.2.7. The fact that this definition is well posed readily descends from the fact that homotopy equivalences induce isometric isomorphisms on bounded cohomology in every degree.
As anticipated above the following deep result of Gromov and Ivanov shows that there exists a isometric isomorphism between the bounded cohomology of any topological space and the bounded cohomology of its fundamental group:
Theorem 0.2.8 ([Iva, Theorem 8.3]). Let X be a path connected topological space. Then, Hb∗(X) is canonically isomorphic to Hb∗(π1(X)).
Remark 0.2.9. The first version of the previous theorem originally appeared in Gromov’s pioneering paper [Gro82, Corollary D and Remark E, page 46] with-out any assumption on the topology of X. Later, Ivanov proved Theorem 0.2.8 only for countable CW complexes (see [Iva87, Theorem 4.1]). However, Ivanov re-cently proved in [Iva, Corollary 6.4] (see Theorem 0.2.4) that bounded cohomology is a weak homotopy invariant. Moreover, in the same paper he extended Theo-rem [Iva87, TheoTheo-rem 4.1] to all CW complexes, hence to all spaces, because every topological space is weakly homotopy equivalent to a CW complex (see e.g. [Hat02, Proposition 4.13]).
One of our goals in this first part of the thesis is to follow Gromov’s original approach via multicomplexes and prove Theorem 0.2.8 for a large class of spaces which contains all CW complexes (without any restriction on the cardinality of their cells). Therefore, thanks to Ivanov’s Theorem 0.2.4, we are able to provide the first complete proof of Theorem 0.2.8 based on the theory of multicomplexes (see Corollaries 4.3.4 and 4.3.5).
We refer the reader to the beginning of Chapter 4 for a broader discussion on the different approaches to Theorem 0.2.8.
Theorem 0.2.8 shows that the bounded cohomology of a space only depends on its fundamental group. This is one reason why it is interesting to study bounded cohomology of groups. The first fundamental property of bounded cohomology is that it vanishes for amenable groups:
Theorem 0.2.10. Let Γ be an amenable group. Then, Hbn(Γ) = 0 ,
for every n > 0. In particular, spaces with amenable fundamental group have van-ishing bounded cohomology in positive degree.
By contrast, bounded cohomology is known to be non zero in degree 2 for neg-atively curved groups as non-Abelian free groups. The first result in that direction has been proved by Brooks and Mitsumatsu in [Bro81] and [Mit84]. Indeed, they showed that the 2-dimensional bounded cohomology of the free group F2 of rank
2 is infinite dimensional. We refer the reader also to an alternative proof due to Rolli [Rol]:
0.2. BOUNDED COHOMOLOGY 7
Theorem 0.2.11 ([Bro81, Mit84], [Rol]). Let F2 be a free group of rank at
least 2. Then, the vector space Hb2(F2) 6= 0 is infinite dimensional.
Remark 0.2.12. Let X be a compact CW complex of dimension n. The theorem above points out an important difference between singular cohomology and bounded cohomology. Indeed, bounded cohomology does not satisfy the dimension axiom, that is we may have Hbk(X) 6= 0 for some k > n.
For example, let X be the wedge sum of 2 circles. Then, since its fundamental group is the free group of rank 2, by applying Theorems 0.2.8 and Theorem 0.2.11, the 2-dimensional bounded cohomology group Hb2(X) is infinite dimensional.
We conclude this paragraph with an extension of Theorem 0.2.8, which under-lines the strong relation between bounded cohomology and amenable groups.
Theorem 0.2.13 (Gromov’s Mapping Theorem ([Gro82, Mapping Theorem, page 40] and [Iva, Theorem 8.4])). Let X, Y be path connected topological spaces and let f : X → Y be a continuous map. Suppose that the induced map f∗: π1(X) →
π1(Y ) is surjective with amenable kernel. Then,
Hb∗(f ) : Hb∗(Y ) → Hb∗(X) is an isometric isomorphism in every positive degree.
Remark 0.2.14. As in the case of Theorem 0.2.8, the previous theorem originally appeared in [Gro82, Mapping Theorem, page 40] without any assumption on X and Y . Later, Ivanov proved in [Iva87, Theorem 4.3] the theorem above for connected countable CW complexes. Moreover, he recently extended in [Iva, Theorem 8.4] his previous result to any path connected CW complexes X and Y and so to any path connected topological space X and Y , thanks again to Theorem 0.2.4.
In the second part of the thesis following Gromov’s approach via multicomplexes and using the invariance of bounded cohomology under weak homotopy equivalences, we provide an alternative proof of Theorem 0.2.13 (see Theorem 5.0.1).
We refer the reader to the beginning of Chapters 4 and 5 for a broader discussion on the different approaches to Gromov’s Mapping Theorem 0.2.13 available in the literature.
We conclude this section with a generalization of Theorem 0.2.10 on the van-ishing of the bounded cohomology of spaces with an amenable fundamental group. We first introduce the fundamental notion of amenable subset of a given topological space:
Definition 0.2.15. Let X be a topological space and let i : Y ,→ X be the inclusion of a subset Y of X. Then Y is amenable (in X) if for every path-connected
component Y0 of Y the image of i∗: π1(Y0) → π1(X) is an amenable subgroup of
π1(X).
Let us now fix a topological space X and let U = {Ui}i∈I be a cover of X,
i.e. suppose that Ui ⊆ X for every i ∈ I and that X = Si∈IUi. We say that the
cover is open if each Ui is open in X, and amenable if each Ui is amenable in X.
The multiplicity of U is defined by
mult(U ) = sup {n | ∃ i1, . . . , in∈ I, ih6= ik for h 6= k, Ui1 ∩ . . . ∩ Uin 6= ∅}
∈ N ∪ {∞} .
The following is the statement due to Gromov describes the vanishing of the bounded cohomology of manifolds admitting some special amenable covers:
Theorem 0.2.16 (Gromov’s Vanishing Theorem [Gro82, Vanishing Theorem, page 40]). Let M be a manifold and let U be an amenable open cover of M . Then, for every n ≥ mult(U ) the comparison map
cn: Hbn(M ) → Hn(X) vanishes.
Gromov’s Vanishing Theorem originally appears in [Gro82, Vanishing Theorem, page 40] with a rather sketchy proof. In the second part of this thesis, we will strengthen this result providing a complete proof which works for any topological space (see Theorem 6.1.2).
Another generalisation and more precise version of Gromov’s Vanishing Theorem is Ivanov’s Cover Theorem. In order to state it, first recall the definition of nerve of a cover:
Definition 0.2.17. To any cover U = {Ui}i∈I of X there is associated a
simpli-cial complex N (U ), called the nerve of the cover, which is defined as follows: the set of vertices of N (U ) is I, and n + 1 elements i0, . . . , in of I span a simplex of N (U )
if and only if
Ui0 ∩ . . . ∩ Uin 6= ∅ .
By definition, mult(U ) = 1 + dim N (U ). We denote by Hn(N (U )) the simplicial cohomology of N with real coefficients.
Following [Iva], we say that a cover U = {Ui}i∈I of a topological space X by
its subsets is nice if either the cover is open and X is hereditary paracompact (i.e. every open subset of X is paracompact), or U is closed and locally finite (i.e. each Ui is closed and each point x ∈ X has a neighbourhood Vx ⊇ {x} which intersects
0.3. SIMPLICIAL VOLUME OF COMPACT MANIFOLDS 9
Theorem 0.2.18 (Ivanov’s Cover Theorem [Iva, Theorem 9.1]). Suppose that U is a nice amenable cover of a topological space X. Let N (U ) be the nerve of U and |N (U )| the geometric realization of N (U ). Then the comparison map c∗: Hb∗(X) → H∗(X) can be factored through the canonical homomorphism H∗(|N (U )|) → H∗(X). Remarks 0.2.19. It is worth mentioning that Ivanov’s Cover Theorem [Iva] holds without any assumption on X if we assume that U is finite (see [Iva, Theo-rem 9.3]).
Moreover, in Section 6.4, we will discuss an alternative proof of Ivanov’s Cover Theorem via multicomplexes under some minor additional hypotheses on U (see Theorem 6.1.1 and Remark 6.1.3).
Finally, we refer the reader to Section 6.1 for an explicit description of the canonical homomorphism mentioned in the statement above.
0.3. Simplicial volume of compact manifolds
The simplicial volume is a homotopy invariant of compact manifolds introduced by Gromov in his seminal paper [Gro82]. It is defined in terms of the `1-norm on singular chains, which is the pre-dual to the `∞-norm on singular cochains defined in the previous section.
Definition 0.3.1. Let X be a topological space. Let C∗(X; R) be the chain
group with coefficient ring R = R, Z.
We endow the chain complex C∗(X; R) with an `1-norm defined as follows: if
Pn i=1αiσi ∈ Ck(X; R), then n X i=1 αiσi 1 = n X i=1 |αi|,
where αi ∈ R and σi: ∆k → X is a singular k-simplex for every i = 1, · · · , n. This
norm descends to an `1-seminorm in homology: let β ∈ Hk(X; R), then
kβk1= inf{kck1| [c] = β, c ∈ Zk(X; R)},
where Zk(X; R) denotes the group of k-cycles of X.
Remark 0.3.2. It is worth noting that k·k1 is just a seminorm in homology.
Indeed, in general kβk1 = 0 does not imply β = 0 ∈ Hk(X; R) (see for instance
Examples 0.3.7).
As for bounded cohomology, any continuous map f : X → Y induces a norm non-increasing map on singular homology
Moreover, homotopic maps induce the same map on singular homology.
Let M be an oriented, connected, closed manifold (i.e. compact without bound-ary) of dimension n. Under this hypothesis the top dimensional integral homology group Hn(M ; Z) is infinite cyclic. We define the integral fundamental class of M ,
[M ]Z∈ Hn(M ; Z), to be the generator of Hn(M ; Z) associated to the orientation of
M . The real fundamental class of M , [M ] ∈ Hn(M ; R), is the image of [M ]Z via
the change of coefficient map Hn(M ; Z) → Hn(M ; R).
Definition 0.3.3. Let M be an oriented, connected, closed n-manifold. The simplicial volume of M is
kM k := k[M ]k1.
Remarks 0.3.4. (i) One may also consider the `1-seminorm of the integral fundamental class of M . Working with integral coefficients, one defines the integral simplicial volume which is a homotopy invariant of oriented closed manifolds but it lacks most of the inheritance properties of the ordinary simplicial volume. For example integral simplicial volume is no more mul-tiplicative with respect to finite coverings (compare with Proposition 0.3.5). However, one can introduce the stable integral simplicial volume which is a variation of the integral simplicial volume which is in fact multiplicative with respect to finite coverings (see for instance [FFM12, FLPS16]). (ii) If M is non-orientable, one may consider the oriented double cover fM of
M and define the simplicial volume of M as follows: kM k := k fM k
2 .
(iii) If M is an oriented, connected and compact manifold M with boundary, one may define the relative simplicial volume to be the `1-seminorm of the (real) relative fundamental class. Since in this thesis we do not analyse the behaviour of compact manifolds with boundary (except for the short Section 4.4), we prefer to postpone to Section 4.4 the formal definition of this invariant. However, most of the following properties of the simplicial volume still hold for the relative simplicial volume. We refer the reader to [Fri17, Section 7] for the corresponding statements in the relative case. 0.3.1. Elementary properties of the simplicial volume. We collect here some elementary properties of the simplicial volume. The following theorem shows that the simplicial volume may provide an obstruction on the possible degrees of maps:
Proposition 0.3.5 ([Gro82]). Let f : M → N be a continuous map between oriented, connected, closed manifolds of the same dimension. Suppose that deg(f ) 6=
0.3. SIMPLICIAL VOLUME OF COMPACT MANIFOLDS 11
0, then
kM k ≥ | deg(f )| · kN k.
Remarks 0.3.6. (i) The proposition above readily implies that the simpli-cial volume is a homotopy invariant. Indeed, since every homotopy equiv-alence f : M → N has degree either 1 or −1 (and the same holds for its homotopy inverse), it is immediate to see that kM k = kN k.
(ii) Another remarkable consequence of the proposition above is the follow-ing: every oriented, connected, closed manifold M which admits a self-map f : M → M of degree | deg(f )| ≥ 2 has zero simplicial volume.
Thanks to Proposition 0.3.5 we are ready to perform the first explicit computa-tion of the simplicial volume.
Examples 0.3.7. Let n ∈ N≥1. Since for any d ∈ Z there exists a map
f : Sn → Sn such that deg(f ) = d, we have that all spheres have zero simplicial
volume:
kSnk = 0.
Analogously, the simplicial volume of the product of a closed connected oriented manifold with any sphere vanishes. For example, the simplicial volume of the n-dimensional torus Tn= S1× · · · S1 vanishes:
kTnk = 0.
One may wonder whether there are situations in which the inequality in Propo-sition 0.3.5 is in fact an equality. This is the case in presence of coverings.
Corollary 0.3.8 ([Gro82]). Let f : M → N be a covering map between ori-ented, connected, closed manifolds. Then,
kM k = | deg(f )| · kN k.
The previous result shows that the simplicial volume is multiplicative with re-spect to finite coverings.
0.3.2. Simplicial volume vs. Riemannian volume. A Riemannian cover-ing between Riemannian manifolds is a topological covercover-ing that is also a local isom-etry. Recall that two Riemannian manifolds M and N are commensurable if there exists a Riemannian manifold H and two finite Riemannian coverings f : H → M and g : H → N . Suppose now that N and M are closed (hence, also H is closed). Then, applying Corollary 0.3.8, we have
kM k | deg(f )| =
kN k | deg(g)|.
Moreover, since the Riemannian volume is multiplicative with respect to finite Rie-mannian coverings, we also have
Vol(M ) | deg(f )| =
Vol(N ) | deg(g)|.
By combining the two equalities above, we get the two commensurable manifolds satisfy:
kM k Vol(M ) =
kN k Vol(N ).
A deep theorem of Gromov shows that the previous formula still holds in the case of Riemannian manifolds with the same universal covering:
Theorem 0.3.9 (Gromov’s Proportionality Principle [Gro82]). Let M be a closed Riemannian manifolds. Then, the ratio
kM k Vol(M )
only depends on the isometry type of the Riemannian universal covering of M . If we restrict our attention to hyperbolic manifolds (that are Riemannian mani-folds whose Riemannian universal covering is isometric to the hyperbolic space Hn), Gromov and Thurston computed the exact value of the previous ratio.
Theorem 0.3.10 ([Thu79, Gro82]). Let M be a hyperbolic manifold. Then, kM k = Vol(M )
vn
,
where vn denotes the maximal volume of geodesic n-simplices in the hyperbolic space
Hn.
Remark 0.3.11. It is worth mentioning that Theorems 0.3.9 and 0.3.10 may be extended to the compactification of complete finite-volume hyperbolic manifolds (see for instance [FP10, FM11, BBI13]). Indeed, Theorem 0.3.9 holds for every oriented, connected, compact Riemannian manifold with boundary. On the con-trary, Theorem 0.3.10 does not hold for compact hyperbolic manifolds with geodesic boundary (see [FP10]).
Theorem 0.3.10 shows that all hyperbolic manifolds have positive simplicial vol-ume. In particular, if M is hyperbolic there cannot exist a self-map f : M → M such that | deg(f )| ≥ 2.
Example 0.3.12. Let us compute the simplicial volume of the oriented, con-nected, closed surface Σg of genus g ≥ 2. Recall that the celebrated Gauss-Bonnet
Theorem implies that the area of closed hyperbolic surfaces satisfies Area(Σg) = −2πχ(Σg).
