aa r X i v : . [ m a t h . G T ] J u l SURVEY ON ASPHERICAL MANIFOLDS
WOLFGANG L ¨UCK
Abstract.
This is a survey on known results and open problems about closedaspherical manifolds, i.e., connected closed manifolds whose universal cover-ings are contractible. Many examples come from certain kinds of non-positivecurvature conditions. The property aspherical, which is a purely homotopytheoretical condition, implies many striking results about the geometry andanalysis of the manifold or its universal covering, and the ring theoretic prop-erties and the K - and L -theory of the group ring associated to its fundamentalgroup. The Borel Conjecture predicts that closed aspherical manifolds aretopologically rigid. The article contains new results about product decompo-sitions of closed aspherical manifolds and an announcement of a result jointwith Arthur Bartels and Shmuel Weinberger about hyperbolic groups withspheres of dimension ≥ Introduction
A space X is called aspherical if it is path connected and all its higher homotopygroups vanish, i.e., π n ( X ) is trivial for n ≥
2. This survey article is devoted toaspherical closed manifolds. These are very interesting objects for many reasons.Often interesting geometric constructions or examples lead to aspherical closedmanifolds. The study of the question which groups occur as fundamental groupsof closed aspherical manifolds is intriguing. The condition aspherical is of purelyhomotopy theoretical nature. Nevertheless there are some interesting questions andconjectures about curvature properties of a closed aspherical Riemann manifold andabout the spectrum of the Laplace operator on its universal covering. The BorelConjecture predicts that aspherical closed topological manifolds are topologicallyrigid and that aspherical compact Poincar´e complexes are homotopy equivalent toclosed manifolds. We discuss the status of some of these questions and conjectures.Examples of exotic aspherical closed manifolds come from hyperbolization tech-niques and we list certain examples. At the end we describe (winking) our universeof closed manifolds.The results about product decompositions of closed aspherical manifolds in Sec-tion 6 are new and Section 8 contains an announcement of a result joint with ArthurBartels and Shmuel Weinberger about hyperbolic groups with spheres of dimension ≥ Date : July 2009.2000
Mathematics Subject Classification.
Key words and phrases. aspherical closed manifolds, topological rigidity, conjectures due toBorel, Novikov, Hopf, Singer, non-positively curved spaces.
The paper is organized as follows:
Contents
0. Introduction 11. Homotopy theory of aspherical manifolds 22. Examples of aspherical manifolds 32.1. Non-positive curvature 32.2. Low-dimensions 32.3. Torsionfree discrete subgroups of almost connected Lie groups 42.4. Hyperbolization 42.5. Exotic aspherical manifolds 43. Non-aspherical closed manifolds 64. The Borel Conjecture 65. Poincar´e duality groups 96. Product decompositions 127. Novikov Conjecture 138. Boundaries of hyperbolic groups 159. L -invariants 169.1. The Hopf and the Singer Conjecture 169.2. L -torsion and aspherical manifolds 169.3. Simplicial volume and L -invariants 179.4. Zero-in-the-Spectrum Conjecture 1710. The universe of closed manifolds 17References 191. Homotopy theory of aspherical manifolds
From the homotopy theory point of view an aspherical CW -complex is com-pletely determined by its fundamental group. Namely Theorem 1.1 (Homotopy classification of aspherical spaces) . (i) Two aspherical CW -complexes are homotopy equivalent if and only if theirfundamental groups are isomorphic;(ii) Let X and Y be connected CW -complexes. Suppose that Y is aspherical.Then we obtain a bijection [ X, Y ] ∼ = −→ [Π( X ) , Π( Y )] , [ f ] [Π( f ))] , where [ X, Y ] is the set of homotopy classes of maps from X to Y , Π( X ) , Π( Y ) are the fundamental groupoids, [Π( X ) , Π( Y )] is the set of naturalequivalence classes of functors from Π( X ) to Π( Y ) and Π( f ) : Π( X ) → Π( Y ) is the functor induced by f : X → Y .Proof. (ii) One easily checks that the map is well-defined. For the proof of surjec-tivity and injectivity one constructs the desired preimage or the desired homotopyinductively over the skeletons of the source.(i) This follows directly from assertion (ii). (cid:3) The description using fundamental groupoids is elegant and base point free,but a reader may prefer its more concrete interpretation in terms of fundamen-tal groups, which we will give next: Choose base points x ∈ X and y ∈ Y .Let hom( π ( X, x ) , π ( Y, y )) be the set of group homomorphisms from π ( X, x )to π ( Y, y ). The group Inn (cid:0) π ( Y, y ) (cid:1) of inner automorphisms of π ( Y, y ) acts on
URVEY ON ASPHERICAL MANIFOLDS 3 hom (cid:0) π ( X, x ) , π ( Y, y ) (cid:1) from the left by composition. We leave it to the reader tocheck that we obtain a bijectionInn (cid:0) π ( Y, y ) (cid:1) \ hom (cid:0) π ( X, x ) , π ( Y, y ) (cid:1) ∼ = −→ [Π( X ) , Π( Y )] , under which the bijection appearing in Lemma 1.1 (ii) sends [ f ] to the class of π ( f, x ) for any choice of representative of f with f ( x ) = y . In the sequel we willoften ignore base points especially when dealing with the fundamental group. Lemma 1.2. A CW -complex X is aspherical if and only if it is connected and itsuniversal covering e X is contractible.Proof. The projection p : e X → X induces isomorphisms on the homotopy groups π n for n ≥ CW -complex is contractible if and only if all itshomotopy groups are trivial (see[99, Theorem IV.7.17 on page 182]. (cid:3) An aspherical CW -complex X with fundamental group π is the same as an Eilenberg Mac-Lane space K ( π, of type ( π,
1) and the same as the classifyingspace Bπ for the group π .2. Examples of aspherical manifolds
In this section we give examples and constructions of aspherical closed manifolds.2.1.
Non-positive curvature.
Let M be a closed smooth manifold. Suppose thatit possesses a Riemannian metric whose sectional curvature is non-positive, i.e.,is ≤ f M inherits a complete Riemannianmetric whose sectional curvature is non-positive. Since f M is simply-connected andhas non-positive sectional curvature, the Hadamard-Cartan Theorem (see [45, 3.87on page 134]) implies that f M is diffeomorphic to R n and hence contractible. Weconclude that f M and hence M is aspherical.2.2. Low-dimensions.
A connected closed 1-dimensional manifold is homeomor-phic to S and hence aspherical.Let M be a connected closed 2-dimensional manifold. Then M is either aspheri-cal or homeomorphic to S or RP . The following statements are equivalent: i.) M is aspherical. ii.) M admits a Riemannian metric which is flat , i.e., with sectionalcurvature constant 0, or which is hyperbolic , i.e., with sectional curvature constant −
1. iii) The universal covering of M is homeomorphic to R .A connected closed 3-manifold M is called prime if for any decomposition asa connected sum M ∼ = M ♯M one of the summands M or M is homeomor-phic to S . It is called irreducible if any embedded sphere S bounds a disk D .Every irreducible closed 3-manifold is prime. A prime closed 3-manifold is eitherirreducible or an S -bundle over S (see [53, Lemma 3.13 on page 28]). A closedorientable 3-manifold is aspherical if and only if it is irreducible and has infinitefundamental group. A closed 3-manifold is aspherical if and only if it is irreducibleand its fundamental group is infinite and contains no element of order 2. Thisfollows from the Sphere Theorem [53, Theorem 4.3 on page 40].Thurston’s Geometrization Conjecture implies that a closed 3-manifold is as-pherical if and only if its universal covering is homeomorphic to R . This followsfrom [53, Theorem 13.4 on page 142] and the fact that the 3-dimensional geome-tries which have compact quotients and whose underlying topological spaces arecontractible have as underlying smooth manifold R (see [89]).A proof of Thurston’s Geometrization Conjecture is given in [74] following ideasof Perelman. WOLFGANG L¨UCK
There are examples of closed orientable 3-manifolds that are aspherical but donot support a Riemannian metric with non-positive sectional curvature (see [66]).For more information about 3-manifolds we refer for instance to [53, 89].2.3.
Torsionfree discrete subgroups of almost connected Lie groups.
Let L be a Lie group with finitely many path components. Let K ⊆ L be a maximalcompact subgroup. Let G ⊆ L be a discrete torsionfree subgroup. Then M = G \ L/K is a closed aspherical manifold with fundamental group G since its universalcovering L/K is diffeomorphic to R n for appropriate n (see [52, Theorem 1. inChapter VI]).2.4. Hyperbolization.
A very important construction of aspherical manifoldscomes from the hyperbolization technique due to Gromov [49]. It turns a cell com-plex into a non-positively curved (and hence aspherical) polyhedron. The roughidea is to define this procedure for simplices such that it is natural under inclusionsof simplices and then define the hyperbolization of a simplicial complex by glu-ing the results for the simplices together as described by the combinatorics of thesimplicial complex. The goal is to achieve that the result shares some of the prop-erties of the simplicial complexes one has started with, but additionally to producea non-positively curved and hence aspherical polyhedron. Since this constructionpreserves local structures, it turns manifolds into manifolds.We briefly explain what the orientable hyperbolization procedure gives. Furtherexpositions of this construction can be found in [19, 22, 24, 25]. We start with afinite-dimensional simplicial complex Σ and a assign to it a cubical cell complex h (Σ) and a natural map c : h (Σ) → Σ with the following properties:(i) h (Σ) is non-positively curved and in particular aspherical;(ii) The natural map c : h (Σ) → Σ induces a surjection on the integral homol-ogy;(iii) π ( f ) : π ( h (Σ)) → π (Σ) is surjective;(iv) If Σ is an orientable manifold, then(a) h (Σ) is a manifold;(b) The natural map c : h (Σ) → Σ has degree one;(c) There is a stable isomorphism between the tangent bundle
T h (Σ) andthe pullback c ∗ T Σ; Remark 2.1 (Characteristic numbers and aspherical manifolds) . Suppose that M is a closed manifold. Then the pullback of the characteristic classes of M under thenatural map c : h ( M ) → M yield the characteristic classes of h ( M ), and M and h ( M ) have the same characteristic numbers. This shows that the condition aspher-ical does not impose any restrictions on the characteristic numbers of a manifold. Remark 2.2 (Bordism and aspherical manifolds) . The conditions above say that c is a normal map in the sense of surgery. One can show that c is normally bordantto the identity map on M . In particular M and h ( M ) are oriented bordant.Consider a bordism theory Ω ∗ for PL-manifolds or smooth manifolds which isgiven by imposing conditions on the stable tangent bundle. Examples are unori-ented bordism, oriented bordism, framed bordism. Then any bordism class can berepresented by an aspherical closed manifold. If two closed aspherical manifoldsrepresent the same bordism class, then one can find an aspherical bordism betweenthem. See [22, Remarks 15.1] and [25, Theorem B].2.5. Exotic aspherical manifolds.
The following result is taken from Davis-Januszkiewicz [25, Theorem 5a.1].
Theorem 2.3.
There is a closed aspherical -manifold N with the following prop-erties: URVEY ON ASPHERICAL MANIFOLDS 5 (i) N is not homotopy equivalent to a P L -manifold;(ii) N is not triangulable, i.e., not homeomorphic to a simplicial complex;(iii) The universal covering e N is not homeomorphic to R ;(iv) N is homotopy equivalent to a piecewise flat, non-positively curved polyhe-dron. The next result is due to Davis-Januszkiewicz [25, Theorem 5a.4].
Theorem 2.4 (Non-PL-example) . For every n ≥ there exists a closed aspherical n -manifold which is not homotopy equivalent to a PL-manifold The proof of the following theorem can be found in [23], [25, Theorem 5b.1].
Theorem 2.5 (Exotic universal covering) . For each n ≥ there exists a closedaspherical n -dimensional manifold such that its universal covering is not homeo-morphic to R n . By the Hadamard-Cartan Theorem (see [45, 3.87 on page 134]) the manifoldappearing in Theorem 2.5 above cannot be homeomorphic to a smooth manifoldwith Riemannian metric with non-positive sectional curvature.The following theorem is proved in [25, Theorem 5c.1 and Remark on page386] by considering the ideal boundary, which is a quasiisometry invariant in thenegatively curved case.
Theorem 2.6 (Exotic example with hyperbolic fundamental group) . For every n ≥ there exists an aspherical closed smooth n -dimensional manifold N whichis homeomorphic to a strictly negatively curved polyhedron and has in particular ahyperbolic fundamental group such that the universal covering is homeomorphic to R n but N is not homeomorphic to a smooth manifold with Riemannian metric withnegative sectional curvature. The next results are due to Belegradek [8, Corollary 5.1], Mess [71] and Wein-berger (see [22, Section 13]).
Theorem 2.7 (Exotic fundamental groups) . (i) For every n ≥ there is a closed aspherical manifold of dimension n whosefundamental group contains an infinite divisible abelian group;(ii) For every n ≥ there is a closed aspherical manifold of dimension n whosefundamental group has an unsolvable word problem and whose simplicialvolume is non-zero. Notice that a finitely presented group with unsolvable word problem is not aCAT(0)-group, not hyperbolic, not automatic, not asynchronously automatic, notresidually finite and not linear over any commutative ring (see [8, Remark 5.2]).The proof of Theorem 2.7 is based on the reflection group trick as it appears forinstance in [22, Sections 8,10 and 13]. It can be summarized as follows.
Theorem 2.8 (Reflection group trick) . Let G be a group which possesses a finitemodel for BG . Then there is a closed aspherical manifold M and a map i : BG → M and r : M → BG such that r ◦ i = id BG . Remark 2.9 (Reflection group trick and various conjectures) . Another interestingimmediate consequence of the reflection group trick is (see also [22, Sections 11])that many well-known conjectures about groups hold for every group which pos-sesses a finite model for BG if and only if it holds for the fundamental group of everyclosed aspherical manifold. This applies for instance to the Kaplansky Conjecture,Unit Conjecture, Zero-divisor-conjecture, Baum-Connes Conjecture, Farrell-JonesConjecture for algebraic K -theory for regular R , Farrell-Jones Conjecture for alge-braic L -theory, the vanishing of e K ( Z G ) and of Wh( G ) = 0, For information about WOLFGANG L¨UCK these conjectures and their links we refer for instance to [6],[68] and [70]. Furthersimilar consequences of the reflection group trick can be found in Belegradek [8].3.
Non-aspherical closed manifolds
A closed manifold of dimension ≥ Lemma 3.1.
The fundamental group of an aspherical finite-dimensional CW -complex X is torsionfree.Proof. Let C ⊆ π ( X ) be a finite cyclic subgroup of π ( X ). We have to showthat C is trivial. Since X is aspherical, C \ e X is a finite-dimensional model for BC .Hence H k ( BC ) = 0 for large k . This implies that C is trivial. (cid:3) Lemma 3.2. If M is a connected sum M ♯M of two closed manifolds M and M of dimension n ≥ which are not homotopy equivalent to a sphere, then M is notaspherical.Proof. We proceed by contradiction. Suppose that M is aspherical. The obvi-ous map f : M ♯M → M ∨ M given by collapsing S n − to a point is ( n − n is the dimension of M and M . Let p : ^ M ∨ M → M ∨ M be the universal covering. By the Seifert-van Kampen Theorem the fundamentalgroup of π ( M ∨ M ) is π ( M ) ∗ π ( M ) and the inclusion of M k → M ∨ M induces injections on the fundamental groups for k = 1 ,
2. We conclude that p − ( M k ) = π ( M ∨ M ) × π ( M k ) g M k for k = 1 ,
2. Since n ≥
3, the map f in-duces an isomorphism on the fundamental groups and an ( n − e f : ^ M ♯M → ^ M ∨ M . Since ^ M ♯M is contractible, H m ( ^ M ∨ M ) = 0 for1 ≤ m ≤ n −
1. Since p − ( M ) ∪ p − ( M ) = ^ M ∨ M and p − ( M ) ∩ p − ( M ) = p − ( {•} ) = π ( M ∨ M ), we conclude H m ( p − ( M k )) = 0 for 1 ≤ m ≤ n − H m ( g M k ) = 0 for 1 ≤ m ≤ n − p − ( M k ) is a disjoint union of copies of g M k .Suppose that π ( M k ) is finite. Since π ( M ♯M ) is torsionfree by Lemma 3.1, π ( M k ) must be trivial and M k = g M k . Since M k is simply connected and H m ( M k ) =0 for 1 ≤ m ≤ n − M k is homotopy equivalent to S n . Since we assume that M k is not homotopy equivalent to a sphere, π ( M k ) is infinite. This implies that themanifold g M k is non-compact and hence H n ( g M k ) = 0. Since g M k is n -dimensional,we conclude H m ( g M k ) = 0 for m ≥
1. Since g M k is simply connected, all ho-motopy groups of g M k vanish by the Hurewicz Theorem [99, Corollary IV.7.8 onpage 180]. We conclude from Lemma 1.2 that M and M are aspherical. Usingthe Mayer-Vietoris argument above one shows analogously that M ∨ M is as-pherical. Since M is by assumption aspherical, M ♯M and M ∨ M are homotopyequivalent by Lemma 1.1 (i). Since they have different Euler characteristics, namely χ ( M ♯M ) = χ ( M ) + χ ( M ) − (1 + ( − n ) and χ ( M ∨ M ) = χ ( M ) + χ ( M ) − (cid:3) The Borel Conjecture
In this section we deal with
Conjecture 4.1 (Borel Conjecture for a group G ) . If M and N are closed as-pherical manifolds of dimensions ≥ with π ( M ) ∼ = π ( N ) ∼ = G , then M and N are homeomorphic and any homotopy equivalence M → N is homotopic to ahomeomorphism. URVEY ON ASPHERICAL MANIFOLDS 7
Definition 4.2 (Topologically rigid) . We call a closed manifold N topologicallyrigid if any homotopy equivalence M → N with a closed manifold M as source ishomotopic to a homeomorphism.If the Borel Conjecture holds for all finitely presented groups, then every closedaspherical manifold is topologically rigid.The main tool to attack the Borel Conjecture is surgery theory and the Farrell-Jones Conjecture. We consider the following special version of the Farrell-JonesConjecture. Conjecture 4.3 (Farrell-Jones Conjecture for torsionfree groups and regular rings) . Let G be a torsionfree group and let R be a regular ring, e.g., a principal idealdomain, a field, or Z . Then(i) K n ( RG ) = 0 for n ≤ − ;(ii) The change of rings homomorphism K ( R ) → K ( RG ) is bijective. (Thisimplies in the case R = Z that the reduced projective class group e K ( Z G ) vanishes;(iii) The obvious map K ( R ) × G/ [ G, G ] → K ( RG ) is surjective. (This impliesin the case R = Z that the Whitehead group Wh( G ) vanishes);(iv) For any orientation homomorphism w : G → {± } the w -twisted L -theoreticassembly map H n ( BG ; w L h−∞i ) ∼ = −→ L h−∞i n ( RG, w ) is bijective. Lemma 4.4.
Suppose that the torsionfree group G satisfies the version of theFarrell-Jones Conjecture stated in Conjecture 4.3 for R = Z .Then the Borel Conjecture is true for closed aspherical manifolds of dimension ≥ with G as fundamental group. Its is true for closed aspherical manifolds ofdimension with G as fundamental group if G is good in the sense of Freedman(see [42] , [43] ).Sketch of the proof. We treat the orientable case only. The topological structureset S top ( M ) of a closed topological manifold M is the set of equivalence classes ofhomotopy equivalences M ′ → M with a topological closed manifold as source and M as target under the equivalence relation, for which f : M → M and f : M → M are equivalent if there is a homeomorphism g : M → M such that f ◦ g and f are homotopic. The Borel Conjecture 4.1 for a group G is equivalent tothe statement that for every closed aspherical manifold M with G ∼ = π ( M ) itstopological structure set S top ( M ) consists of a single element, namely, the class ofid : M → M .The surgery sequence of a closed orientable topological manifold M of dimension n ≥ . . . → N n +1 (cid:0) M × [0 , , M × { , } (cid:1) σ −→ L sn +1 (cid:0) Z π ( M ) (cid:1) ∂ −→ S top ( M ) η −→ N n ( M ) σ −→ L sn (cid:0) Z π ( M ) (cid:1) , which extends infinitely to the left. It is the basic tool for the classification oftopological manifolds. (There is also a smooth version of it.) The map σ ap-pearing in the sequence sends a normal map of degree one to its surgery ob-struction. This map can be identified with the version of the L -theory assemblymap where one works with the 1-connected cover L s ( Z ) h i of L s ( Z ). The map H k (cid:0) M ; L s ( Z ) h i (cid:1) → H k (cid:0) M ; L s ( Z ) (cid:1) is injective for k = n and an isomorphism for k > n . Because of the K -theoretic assumptions we can replace the s -decoration withthe h−∞i -decoration. Therefore the Farrell-Jones Conjecture implies that the maps WOLFGANG L¨UCK σ : N n ( M ) → L sn (cid:0) Z π ( M ) (cid:1) and N n +1 (cid:0) M × [0 , , M × { , } (cid:1) σ −→ L sn +1 (cid:0) Z π ( M ) (cid:1) are injective respectively bijective and thus by the surgery sequence that S top ( M )is a point and hence the Borel Conjecture 4.1 holds for M . More details can befound e.g., in [39, pages 17,18,28], [87, Chapter 18]. (cid:3) Remark 4.5 (The Borel Conjecture in low dimensions) . The Borel Conjecture istrue in dimension ≤ Remark 4.6 (Topological rigidity for non-aspherical manifolds) . Topological rigid-ity phenomenons do hold also for some non-aspherical closed manifolds. For in-stance the sphere S n is topologically rigid by the Poincar´e Conjecture. The Poincar´eConjecture is known to be true in all dimensions. This follows in high dimensionsfrom the h -cobordism theorem, in dimension four from the work of Freedman [42],in dimension three from the work of Perelman as explained in [62, 73] and and indimension two from the classification of surfaces.Many more examples of classes of manifolds which are topologically rigid aregiven and analyzed in Kreck-L¨uck [65]. For instance the connected sum of closedmanifolds of dimension ≥ S k × S n is topologically rigid if and only if k and n are odd. An inte-gral homology sphere of dimension n ≥ Z → Z [ π ( M )] induces an isomorphism of simple L -groups L sn +1 ( Z ) → L sn +1 (cid:0) Z [ π ( M )] (cid:1) . Remark 4.7 (The Borel Conjecture does not hold in the smooth category) . TheBorel Conjecture 4.1 is false in the smooth category, i.e., if one replaces topolog-ical manifold by smooth manifold and homeomorphism by diffeomorphism. Thetorus T n for n ≥ Remark 4.8 (The Borel Conjecture versus Mostow rigidity) . The examples ofFarrell-Jones [31, Theorem 0.1] give actually more. Namely, it yields for given ǫ > M whose sectional curvature lies in theinterval [1 − ǫ, − ǫ ] and a closed hyperbolic manifold M such that M and M are homeomorphic but no diffeomorphic. The idea of the construction is essentiallyto take the connected sum of M with exotic spheres. Notice that by definition M were hyperbolic if we would take ǫ = 0. Hence this example is remarkable in viewof Mostow rigidity , which predicts for two closed hyperbolic manifolds N and N that they are isometrically diffeomorphic if and only if π ( N ) ∼ = π ( N ) and anyhomotopy equivalence N → N is homotopic to an isometric diffeomorphism.One may view the Borel Conjecture as the topological version of Mostow rigidity.The conclusion in the Borel Conjecture is weaker, one gets only homeomorphismsand not isometric diffeomorphisms, but the assumption is also weaker, since thereare many more aspherical closed topological manifolds than hyperbolic closed man-ifolds. Remark 4.9 (The work of Farrell-Jones) . Farrell-Jones have made deep contri-butions to the Borel Conjecture. They have proved it in dimension ≥ URVEY ON ASPHERICAL MANIFOLDS 9 manifolds and for closed aspherical manifolds whose fundamental group is isomor-phic to the fundamental group of a complete non-positively curved Riemannianmanifold which is A-regular (see [32, 33, 35, 36]).The following result is due to Bartels and L¨uck [4].
Theorem 4.10.
Let C be the smallest class of groups satisfying: • Every hyperbolic group belongs to C ; • Every group that acts properly, isometrically and cocompactly on a com-plete proper
CAT(0) -space belongs to C ; • If G and G belong to C , then both G ∗ G and G × G belong to C ; • If H is a subgroup of G and G ∈ C , then H ∈ C ; • Let { G i | i ∈ I } be a directed system of groups (with not necessarily injec-tive structure maps) such that G i ∈ C for every i ∈ I . Then the directedcolimit colim i ∈ I G i belongs to C .Then every group G in C satisfies the version of the Farrell-Jones Conjecturestated in Conjecture 4.3. Remark 4.11 (Exotic closed aspherical manifolds) . Theorem 4.10 implies that theexotic aspherical manifolds mentioned in Subsection 2.5 satisfy the Borel Conjecturein dimension ≥ Remark 4.12 (Directed colimits of hyperbolic groups) . There are also a variety ofinteresting groups such as lacunary groups in the sense of Olshanskii-Osin-Sapir [80]or groups with expanders as they appear in the counterexample to the
Baum-ConnesConjecture with coefficients due to Higson-Lafforgue-Skandalis [54] and which havebeen constructed by Arzhantseva-Delzant [2, Theorem 7.11 and Theorem 7.12].Since these arise as colimits of directed systems of hyperbolic groups, they dosatisfy the Farrell-Jones Conjecture and the Borel Conjecture in dimension ≥ Bost Conjecture has also been proved for colimits of hyperbolic groups byBartels-Echterhoff-L¨uck [3].The original source for the (Fibered) Farrell-Jones Conjecture is the paper byFarrell-Jones [34, 1.6 on page 257 and 1.7 on page 262]. The C ∗ -analogue of theFarrell-Jones Conjecture is the Baum-Connes Conjecture whose formulation can befound in [7, Conjecture 3.15 on page 254]. For more information about the Baum-Connes Conjecture and the Farrell-Jones Conjecture and literature about them werefer for instance to the survey article [70].5. Poincar´e duality groups
The following definition is due to Johnson-Wall [59].
Definition 5.1 (Poincar´e duality group) . A group G is called a Poincar´e dualitygroup of dimension n if the following conditions holds:(i) The group G is of type FP, i.e., the trivial Z G -module Z possesses a finite-dimensional projective Z G -resolution by finitely generated projective Z G -modules;(ii) We get an isomorphism of abelian groups H i ( G ; Z G ) ∼ = (cid:26) { } for i = n ; Z for i = n. The next definition is due to Wall [96]. Recall that a CW -complex X is called finitely dominated if there exists a finite CW -complex Y and maps i : X → Y and r : Y → X with r ◦ i ≃ id X . Definition 5.2 (Poincar´e complex) . Let X be a finitely dominated connected CW -complex with fundamental group π .It is called a Poincar´e complex of dimension n if there exists an orientationhomomorphism w : π → {± } and an element[ X ] ∈ H πn ( e X ; w Z ) = H n (cid:0) C ∗ ( e X ) ⊗ Z π w Z (cid:1) in the n -th π -equivariant homology of its universal covering e X with coefficients inthe Z G -module w Z , such that the up to Z π -chain homotopy equivalence unique Z π -chain map − ∩ [ X ] : C n −∗ ( e X ) = hom Z π (cid:0) C n −∗ ( e X ) , Z π (cid:1) → C ∗ ( e X )is a Z π -chain homotopy equivalence. Here w Z is the Z G -module, whose underlyingabelian group is Z and on which g ∈ π acts by multiplication with w ( g ).If in addition X is a finite CW -complex, we call X a finite Poincar´e dualitycomplex of dimension n .A topological space X is called an absolute neighborhood retract or briefly ANRif for every normal space Z , every closed subset Y ⊆ Z and every (continuous)map f : Y → X there exists an open neighborhood U of Y in Z together withan extension F : U → Z of f to U . A compact n -dimensional homology ANR -manifold X is a compact absolute neighborhood retract such that it has a countablebasis for its topology, has finite topological dimension and for every x ∈ X theabelian group H i ( X, X − { x } ) is trivial for i = n and infinite cyclic for i = n . Aclosed n -dimensional topological manifold is an example of a compact n -dimensionalhomology ANR-manifold (see [21, Corollary 1A in V.26 page 191]). Theorem 5.3 (Homology ANR-manifolds and finite Poincar´e complexes) . Let M be a closed topological manifold, or more generally, a compact homology ANR -manifold of dimension n . Then M is homotopy equivalent to a finite n -dimensionalPoincar´e complex.Proof. A closed topological manifold, and more generally a compact ANR, has thehomotopy type of a finite CW -complex (see [61, Theorem 2.2]. [98]). The usualproof of Poincar´e duality for closed manifolds carries over to homology manifolds. (cid:3) Theorem 5.4 (Poincar´e duality groups) . Let G be a group and n ≥ be an integer.Then:(i) The following assertions are equivalent:(a) G is finitely presented and a Poincar´e duality group of dimension n ;(b) There exists an n -dimensional aspherical Poincar´e complex with G asfundamental group;(ii) Suppose that e K ( Z G ) = 0 . Then the following assertions are equivalent:(a) G is finitely presented and a Poincar´e duality group of dimension n ;(b) There exists a finite n -dimensional aspherical Poincar´e complex with G as fundamental group;(iii) A group G is a Poincar´e duality group of dimension if and only if G ∼ = Z ;(iv) A group G is a Poincar´e duality group of dimension if and only if G isisomorphic to the fundamental group of a closed aspherical surface;Proof. (i) Every finitely dominated CW -complex has a finitely presented funda-mental group since every finite CW -complex has a finitely presented group and agroup which is a retract of a finitely presented group is again finitely presented [94,Lemma 1.3]. If there exists a CW -model for BG of dimension n , then the cohomo-logical dimension of G satisfies cd( G ) ≤ n and the converse is true provided that URVEY ON ASPHERICAL MANIFOLDS 11 n ≥ ⇒ (i)a holds for all n ≥ ⇒ (i)b holds for n ≥
3. For more details we refer to [59, Theorem 1].The remaining part to show the implication (i)a = ⇒ (i)b for n = 1 , n ≥ CW -complex ishomotopy equivalent to a finite CW -complex and takes values in e K ( Z π ) (see [37,72, 94, 95]). The implication (ii)b = ⇒ (ii)a holds for all n ≥
1. The remainingpart to show the implication (ii)a = ⇒ (ii)b holds follows from assertions (iii)and (iv).(iii) Since S = B Z is a 1-dimensional closed manifold, Z is a finite Poincare dualitygroup of dimension 1 by Theorem 5.3. We conclude from the (easy) implication(i)b = ⇒ (i)a appearing in assertion (i) that Z is a Poincar´e duality group ofdimension 1. Suppose that G is a Poincar´e duality group of dimension 1. Since thecohomological dimension of G is 1, it has to be a free group (see [91, 92]). Since thehomology group of a group of type FP is finitely generated, G is isomorphic to afinitely generated free group F r of rank r . Since H ( BF r ) ∼ = Z r and H ( BF r ) ∼ = Z ,Poincar´e duality can only hold for r = 1, i.e., G is Z .(iv) This is proved in [27, Theorem 2]. See also [10, 11, 26, 28]. (cid:3) Conjecture 5.5 (Aspherical Poincar´e complexes) . Every finite Poincar´e complexis homotopy equivalent to a closed manifold.
Conjecture 5.6 (Poincare duality groups) . A finitely presented group is a n -dimensional Poincar´e duality group if and only if it is the fundamental group ofa closed n -dimensional topological manifold. Because of Theorem 5.3 and Theorem 5.4 (i), Conjecture 5.5 and Conjecture 5.6are equivalent.The disjoint disk property says that for any ǫ > f, g : D → M thereare maps f ′ , g ′ : D → M so that the distance between f and f ′ and the distancebetween g and g ′ are bounded by ǫ and f ′ ( D ) ∩ g ′ ( D ) = ∅ . Lemma 5.7.
Suppose that the torsionfree group G and the ring R = Z satisfy theversion of the Farrell-Jones Conjecture stated in Theorem 4.3. Let X be a Poincar´ecomplex of dimension ≥ with π ( X ) ∼ = G . Then X is homotopy equivalent to acompact homology ANR -manifold satisfying the disjoint disk property.Proof.
See [87, Remark 25.13 on page 297], [15, Main Theorem on page 439 andSection 8] and [16, Theorem A and Theorem B]. (cid:3)
Remark 5.8 (Compact homology ANR-manifolds versus closed topological mani-folds) . In the following all manifolds have dimension ≥
6. One would prefer if in theconclusion of Lemma 5.7 one could replace “compact homology ANR-manifold” by“closed topological manifold”. The problem is that in the geometric exact surgerysequence one has to work with the 1-connective cover L h i of the L -theory spectrum L , whereas in the assembly map appearing in the Farrell-Jones setting one uses the L -theory spectrum L . The L -theory spectrum L is 4-periodic, i.e., π n ( L ) ∼ = π n +4 ( L )for n ∈ Z . The 1-connective cover L h i comes with a map of spectra f : L h i → L such that π n ( f ) is an isomorphism for n ≥ π n ( L h i ) = 0 for n ≤
0. Since π ( L ) ∼ = Z , one misses a part involving L ( Z ) of the so called total surgery ob-struction due to Ranicki, i.e., the obstruction for a finite Poincar´e complex to behomotopy equivalent to a closed topological manifold, if one deals with the peri-odic L -theory spectrum L and picks up only the obstruction for a finite Poincar´e complex to be homotopy equivalent to a compact homology ANR-manifold, theso called four-periodic total surgery obstruction . The difference of these two ob-structions is related to the resolution obstruction of Quinn which takes values in L ( Z ). Any element of L ( Z ) can be realized by an appropriate compact homologyANR-manifold as its resolution obstruction . There are compact homology ANR-manifolds that are not homotopy equivalent to closed manifolds. But no example ofan aspherical compact homology ANR-manifold that is not homotopy equivalent toa closed topological manifold is known. For an aspherical compact homology ANR-manifold M , the total surgery obstruction and the resolution obstruction carry thesame information. So we could replace in the conclusion of Lemma 5.7 “compacthomology ANR-manifold” by “closed topological manifold” if and only if every as-pherical compact homology ANR-manifold with the disjoint disk property admitsa resolution.We refer for instance to [15, 38, 85, 86, 87] for more information about this topic. Question 5.9 (Vanishing of the resolution obstruction in the aspherical case) . Isevery aspherical compact homology ANR-manifold homotopy equivalent to a closedmanifold? 6.
Product decompositions
In this section we show that, roughly speaking, a closed aspherical manifold M is a product M × M if and only if its fundamental group is a product π ( M ) = G × G and that such a decomposition is unique up to homeomorphism. Theorem 6.1 (Product decomposition) . Let M be a closed aspherical manifoldof dimension n with fundamental group G = π ( M ) . Suppose we have a productdecomposition p × p : G ∼ = −→ G × G . Suppose that G , G and G satisfy the version of the Farrell-Jones Conjecture statedin Theorem 4.3 in the case R = Z .Then G , G and G are Poincar´e duality groups whose cohomological dimensionssatisfy n = cd( G ) = cd( G ) + cd( G ) . Suppose in the sequel: • the cohomological dimension cd( G i ) is different from , and for i = 1 , . • n ≥ or n ≤ or ( n = 4 and G is good in the sense of Freedmann);Then:(i) There are topological closed aspherical manifolds M and M together withisomorphisms v i : π ( M i ) ∼ = −→ G i and maps f i : M → M i for i = 1 , such that f = f × f : M → M × M is a homeomorphism and v i ◦ π ( f i ) = p i (up to inner automorphisms) for i = 1 , ;(ii) Suppose we have another such choice of topological closed aspherical man-ifolds M ′ and M ′ together with isomorphisms v ′ i : π ( M ′ i ) ∼ = −→ G i URVEY ON ASPHERICAL MANIFOLDS 13 and maps f ′ i : M → M ′ i for i = 1 , such that the map f ′ = f ′ × f ′ is a homotopy equivalence and v ′ i ◦ π ( f ′ i ) = p i (up to inner automorphisms) for i = 1 , . Then there arefor i = 1 , homeomorphisms h i : M i → M ′ i such that h i ◦ f i ≃ f ′ i and v i ◦ π ( h i ) = v ′ i holds for i = 1 , .Proof. In the sequel we identify G = G × G by p × p . Since the closed manifold M is a model for BG and cd( G ) = n , we can choose BG to be an n -dimensionalfinite Poincar´e complex in the sense of Definition 5.2 by Theorem 5.3.From BG = B ( G × G ) ≃ BG × BG we conclude that there are finitelydominated CW -models for BG i for i = 1 ,
2. Since e K ( Z G i ) vanishes for i = 0 , BG i of dimension max { cd( G i ) , } . Weconclude from [47], [84] that BG and BG are Poincar´e complexes. One easilychecks using the K¨unneth formula that n = cd( G ) = cd( G ) + cd( G ) . If cd( G i ) = 1, then BG i is homotopy equivalent to a manifold, namely S , byTheorem 5.4 (iii). If cd( G i ) = 2, then BG i is homotopy equivalent to a manifoldby Theorem 5.4 (iv). Hence it suffices to show for i = 1 , BG i is homotopyequivalent to a closed aspherical manifold, provided that cd( G i ) ≥ G i satisfies the version of the Farrell-Jones Conjecturestated in Theorem 4.3 in the case R = Z , there exists a compact homology ANR-manifold M i that satisfies the disjoint disk property and is homotopy equivalentto BG i (see Lemma 5.7). Hence it remains to show that Quinn’s resolution ob-struction I ( M i ) ∈ (1 + 8 · Z ) is 1 (see [86, Theorem 1.1]). Since this obstructionis multiplicative (see [86, Theorem 1.1]), we get I ( M × M ) = I ( M ) · I ( M ). Ingeneral the resolution obstruction is not a homotopy invariant, but it is known tobe a homotopy invariant for aspherical compact ANR-manifolds if the fundamentalgroup satisfies the Novikov Conjecture 7.2 (see [15, Proposition on page 437]). Since G i satisfies the version of the Farrell-Jones Conjecture stated in Theorem 4.3 in thecase R = Z , it satisfies the Novikov Conjecture by Lemma 4.4 and Remark 7.4.Hence I ( M × M ) = I ( M ). Since I ( M ) is a closed manifold, we have I ( M ) = 1.Hence I ( M i ) = 1 and M i is homotopy equivalent to a closed manifold. This finishesthe proof of assertion (i).Assertion (ii) follows from Lemma 4.4. (cid:3) Remark 6.2 (Product decompositions and non-positive sectional curvature) . Thefollowing result has been proved by Gromoll-Wolf [48, Theorem 2]. Let M be aclosed Riemannian manifold with non-positive sectional curvature. Suppose thatwe are given a splitting of its fundamental group π ( M ) = G × G and that thecenter of π ( M ) is trivial. Then this splitting comes from an isometric productdecomposition of closed Riemannian manifolds of non-positive sectional curvature M = M × M . 7. Novikov Conjecture
Let G be a group and let u : M → BG be a map from a closed oriented smoothmanifold M to BG . Let L ( M ) ∈ M k ∈ Z ,k ≥ H k ( M ; Q )be the L -class of M . Its k -th entry L ( M ) k ∈ H k ( M ; Q ) is a certain homogeneouspolynomial of degree k in the rational Pontrjagin classes p i ( M ; Q ) ∈ H i ( M ; Q ) for i = 1 , , . . . , k such that the coefficient s k of the monomial p k ( M ; Q ) is differentfrom zero. The L -class L ( M ) is determined by all the rational Pontrjagin classesand vice versa. The L -class depends on the tangent bundle and thus on the differ-entiable structure of M . For x ∈ Q k ≥ H k ( BG ; Q ) define the higher signature of M associated to x and u to be the integersign x ( M, u ) := hL ( M ) ∪ f ∗ x, [ M ] i . (7.1)We say that sign x for x ∈ H ∗ ( BG ; Q ) is homotopy invariant if for two closedoriented smooth manifolds M and N with reference maps u : M → BG and v : N → BG we have sign x ( M, u ) = sign x ( N, v ) , whenever there is an orientation preserving homotopy equivalence f : M → N suchthat v ◦ f and u are homotopic. If x = 1 ∈ H ( BG ), then the higher signaturesign x ( M, u ) is by the Hirzebruch signature formula (see [56, 57]) the signature of M itself and hence an invariant of the oriented homotopy type. This is one motivationfor the following conjecture. Conjecture 7.2 (Novikov Conjecture) . Let G be a group. Then sign x is homotopyinvariant for all x ∈ Q k ∈ Z ,k ≥ H k ( BG ; Q ) . This conjecture appears for the first time in the paper by Novikov [78, § Theorem 7.3 (Topological invariance of rational Pontrjagin classes) . The rationalPontrjagin classes p k ( M, Q ) ∈ H k ( M ; Q ) are topological invariants, i.e. for ahomeomorphism f : M → N of closed smooth manifolds we have H k ( f ; Q ) (cid:0) p k ( M ; Q ) (cid:1) = p k ( N ; Q ) for all k ≥ and in particular H ∗ ( f ; Q )( L ( M )) = L ( N ) . The rational Pontrjagin classes are not homotopy invariants and the integralPontrjagin classes p k ( M ) are not homeomorphism invariants (see for instance [64,Example 1.6 and Theorem 4.8]). Remark 7.4 (The Novikov Conjecture and aspherical manifolds) . Let f : M → N be a homotopy equivalence of closed aspherical manifolds. Suppose that the BorelConjecture 4.1 is true for G = π ( N ). This implies that f is homotopic to ahomeomorphism and hence by Theorem 7.3 f ∗ ( L ( M )) = L ( N ) . But this is equivalent to the conclusion of the Novikov Conjecture in the case N = BG . Conjecture 7.5.
A closed aspherical smooth manifold does not admit a Riemann-ian metric of positive scalar curvature.
Proposition 7.6.
Suppose that the strong Novikov Conjecture is true for thegroup G , i.e., the assembly map K n ( BG ) → K n ( C ∗ r ( G )) is rationally injective for all n ∈ Z . Let M be a closed aspherical smooth manifoldwhose fundamental group is isomorphic to G .Then M carries no Riemannian metric of positive scalar curvature.Proof. See [88, Theorem 3.5]. (cid:3)
URVEY ON ASPHERICAL MANIFOLDS 15
Proposition 7.7.
Let G be a group. Suppose that the assembly map K n ( BG ) → K n ( C ∗ r ( G )) is rationally injective for all n ∈ Z . Let M be a closed aspherical smooth manifoldwhose fundamental group is isomorphic to G .Then M satisfies the Zero-in-the-Spectrum Conjecture 9.5Proof. See [67, Corollary 4]. (cid:3)
We refer to [70, Section 5.1.3] for a discussion about the large class of groupsfor which the assembly map K n ( BG ) → K n ( C ∗ r ( G )) is known to be injective orrationally injective. 8. Boundaries of hyperbolic groups
We announce the following two theorems joint with Arthur Bartels and ShmuelWeinberger. For the notion of the boundary of a hyperbolic group and its mainproperties we refer for instance to [60].
Theorem 8.1.
Let G be a torsion-free hyperbolic group and let n be an integer ≥ . Then:(i) The following statements are equivalent:(a) The boundary ∂G is homeomorphic to S n − ;(b) There is a closed aspherical topological manifold M such that G ∼ = π ( M ) , its universal covering f M is homeomorphic to R n and the com-pactification of f M by ∂G is homeomorphic to D n ;(ii) The aspherical manifold M appearing in the assertion above is unique upto homeomorphism. The proof depends strongly on the surgery theory for compact homology ANR-manifolds due to Bryant-Ferry-Mio-Weinberger [15] and the validity of the K - and L -theoretic Farrell-Jones Conjecture for hyperbolic groups due to Bartels-Reich-L¨uck [5] and Bartels-L¨uck [4]. It seems likely that this result holds also if n = 5.Our methods can be extended to this case if the surgery theory from [15] can beextended to the case of 5-dimensional compact homology ANR-manifolds.We do not get information in dimensions n ≤ S , since virtually cyclic groups are the only hyper-bolic groups which are known to be good in the sense of Friedman [43]. In the case n = 3 there is the conjecture of Cannon [17] that a group G acts properly, isomet-rically and cocompactly on the 3-dimensional hyperbolic plane H if and only if itis a hyperbolic group whose boundary is homeomorphic to S . Provided that theinfinite hyperbolic group G occurs as the fundamental group of a closed irreducible3-manifold, Bestvina-Mess [9, Theorem 4.1] have shown that its universal coveringis homeomorphic to R and its compactification by ∂G is homeomorphic to D ,and the Geometrization Conjecture of Thurston implies that M is hyperbolic and G satisfies Cannon’s conjecture. The problem is solved in the case n = 2, namely,for a hyperbolic group G its boundary ∂G is homeomorphic to S if and only if G is a Fuchsian group (see [18, 41, 44]).For every n ≥ n whose fundamental group G is hyperbolic, which is homeomorphic to a closedaspherical smooth manifold and whose universal covering is homeomorphic to R n ,but the boundary ∂G is not homeomorphic to S n − , see [25, Theorem 5c.1 onpage 384 and Remark on page 386]. Thus the condition that ∂G is a sphere for atorsion-free hyperbolic group is (in high dimensions) not equivalent to the existenceof an aspherical manifold whose fundamental group is G . Theorem 8.2.
Let G be a torsion-free hyperbolic group and let n be an integer ≥ . Then(i) The following statements are equivalent:(a) The boundary ∂G has the integral ˇCech cohomology of S n − ;(b) G is a Poincar´e duality group of dimension n ;(c) There exists a compact homology ANR -manifold M homotopy equiv-alent to BG . In particular, M is aspherical and π ( M ) ∼ = G ;(ii) If the statements in assertion (i) hold, then the compact homology ANR -manifold M appearing there is unique up to s -cobordism of compact ANR -homology manifolds.
The discussion of compact homology ANR-manifolds versus closed topologicalmanifolds of Remark 5.8 and Question 5.9 are relevant for Theorem 8.2 as well.In general the boundary of a hyperbolic group is not locally a Euclidean spacebut has a fractal behavior. If the boundary ∂G of an infinite hyperbolic group G contains an open subset homeomorphic to Euclidean n -space, then it is homeomor-phic to S n . This is proved in [60, Theorem 4.4], where more information about theboundaries of hyperbolic groups can be found.9. L -invariants Next we mention some prominent conjectures about aspherical manifolds and L -invariants. For more information about these conjectures and their status werefer to [68] and [69].9.1. The Hopf and the Singer Conjecture.Conjecture 9.1 (Hopf Conjecture) . If M is an aspherical closed manifold of evendimension, then ( − dim( M ) / · χ ( M ) ≥ . If M is a closed Riemannian manifold of even dimension with sectional curvature sec( M ) , then ( − dim( M ) / · χ ( M ) > if sec( M ) < − dim( M ) / · χ ( M ) ≥ if sec( M ) ≤ χ ( M ) = 0 if sec( M ) = 0; χ ( M ) ≥ if sec( M ) ≥ χ ( M ) > if sec( M ) > . Conjecture 9.2 (Singer Conjecture) . If M is an aspherical closed manifold, then b (2) p ( f M ) = 0 if p = dim( M ) . If M is a closed connected Riemannian manifold with negative sectional curvature,then b (2) p ( f M ) (cid:26) = 0 if p = dim( M ); > if p = dim( M ) . L -torsion and aspherical manifolds.Conjecture 9.3 ( L -torsion for aspherical manifolds) . If M is an aspherical closedmanifold of odd dimension, then f M is det - L -acyclic and ( − dim( M ) − · ρ (2) ( f M ) ≥ . If M is a closed connected Riemannian manifold of odd dimension with negativesectional curvature, then f M is det - L -acyclic and ( − dim( M ) − · ρ (2) ( f M ) > . URVEY ON ASPHERICAL MANIFOLDS 17 If M is an aspherical closed manifold whose fundamental group contains an amenableinfinite normal subgroup, then f M is det - L -acyclic and ρ (2) ( f M ) = 0 . Simplicial volume and L -invariants.Conjecture 9.4 (Simplicial volume and L -invariants) . Let M be an asphericalclosed orientable manifold. Suppose that its simplicial volume || M || vanishes. Then f M is of determinant class and b (2) p ( f M ) = 0 for p ≥ ρ (2) ( f M ) = 0 . Zero-in-the-Spectrum Conjecture.Conjecture 9.5 (Zero-in-the-spectrum Conjecture) . Let f M be a complete Rie-mannian manifold. Suppose that f M is the universal covering of an aspherical closedRiemannian manifold M (with the Riemannian metric coming from M ). Then forsome p ≥ zero is in the Spectrum of the minimal closure (∆ p ) min : dom (cid:0) (∆ p ) min (cid:1) ⊂ L Ω p ( f M ) → L Ω p ( f M ) of the Laplacian acting on smooth p -forms on f M . Remark 9.6 (Non-aspherical counterexamples to the Zero-in-the-Spectrum Con-jecture) . For all of the conjectures about aspherical spaces stated in this article itis obvious that they cannot be true if one drops the condition aspherical except forthe zero-in-the-Spectrum Conjecture 9.5. Farber and Weinberger [30] gave the firstexample of a closed Riemannian manifold for which zero is not in the spectrumof the minimal closure (∆ p ) min : dom ((∆ p ) min ) ⊂ L Ω p ( f M ) → L Ω p ( f M ) of theLaplacian acting on smooth p -forms on f M for each p ≥
0. The construction byHigson, Roe and Schick [55] yields a plenty of such counterexamples. But there areno aspherical counterexamples known.10.
The universe of closed manifolds
At the end we describe (winking) our universe of closed manifolds.The idea of a random group has successfully been used to construct groups withcertain properties, see for instance [2], [46], [50, 9.B on pages273ff], [51], [79],[82],[90] and [100]. In a precise statistical sense almost all finitely presented groupsare hyperbolic see [81]. One can actually show that in a precise statistical sensealmost all finitely presented groups are torsionfree hyperbolic and in particular havea finite model for their classifying space. In most cases it is given by the limit for n → ∞ of the quotient of the number of finitely presented groups with a certainproperty (P) which are given by a presentation satisfying a certain condition C n by the number of all finitely presented groups which are given by a presentationsatisfying condition C n .It is not clear what it means in a precise sense to talk about a random closedmanifold. Nevertheless, the author’s intuition is that almost all closed manifoldsare aspherical. (A related question would be whether a random closed smoothmanifold admits a Riemannian metric with non-positive sectional curvature.) Thisintuition is supported by Remark 2.1. It is certainly true in dimension 2 since onlyfinitely many closed surfaces are not aspherical. The characterization of closed3-dimensional manifolds in Subsection 2.2 seems to fit as well. In the sequel weassume that this (vague) intuition is correct. If we combine these considerations, we get that almost all closed manifolds areaspherical and have a hyperbolic fundamental group. Since except in dimension 4the Borel Conjecture is known in this case by Lemma 4.4, Remark 4.5 and The-orem 4.10, we get as a consequence that almost almost all closed manifolds areaspherical and topologically rigid.A closed manifold M is called asymmetric if every finite group which acts effec-tively on M is trivial. This is equivalent to the statement that for any choice ofRiemannian metric on M the group of isometries is trivial (see [63, Introduction]).A survey on asymmetric closed manifolds can be found in [83]. The first con-structions of asymmetric closed aspherical manifolds are due to Connor-Raymond-Weinberger [20]. The first simply-connected asymmetric manifold has been con-structed by Kreck [63] answering a question of Raymond and Schultz [13, page 260]which was repeated by Adem and Davis [1] in their problem list. Raymond andSchultz expressed also their feeling that a random manifold should be asymmetric.Borel has shown that an aspherical closed manifold is asymmetric if its fundamen-tal group is centerless and its outer automorphism group is torsionfree (see themanuscript “On periodic maps of certain K ( π, n -dimensional standard sphere S n can be characterized among (simply connected) closed Riemannian manifolds ofdimension n by the property that its isometry group has maximal dimension. Moreprecisely, if M is a closed n -dimensional smooth manifold, then the dimension ofits isometry group for any Riemannian metric is bounded by n ( n + 1) / n ( n + 1) / M is diffeomorphic to S n or RP n ;see Hsiang [58], where the Ph.D-thesis of Eisenhart is cited and the dimension ofthe isometry group of exotic spheres is investigated. It is likely that the humantaste whether a geometric object is beautiful is closely related to the question howmany symmetries it admits. In general it seems to be the case that a humanbeing is attracted by unusual representatives among mathematical objects such asgroups or closed manifolds and not by the generic ones. In group theory it is clearthat random groups can have very strange properties and that these groups are tosome extend scary. The analogous statement seems to hold for closed topologicalmanifolds.At the time of writing the author cannot really name a group which could bea potential counterexample to the Farrell-Jones Conjecture or other conjecturesdiscussed in this article. But the author has the feeling that nevertheless the class ofgroups, for which we can prove the conjecture and which is for “human standards”quite large, is only a very tiny portion of the whole universe of groups and thequestion whether these conjectures are true for all groups is completely open.Here is an interesting parallel to our actual universe. If you materialize at arandom point in the universe it will be very cold and nothing will be there. There URVEY ON ASPHERICAL MANIFOLDS 19 is no interaction between different random points, i.e., it is rigid. A human beingwill not like this place, actually even worse, it cannot exist at such a random place.But there are unusual rare non-generic points in the universe, where human beingscan exist such as the surface of our planet and there a lot of things and interactionsare happening. And human beings tend to think that the rest of the universelooks like the place they are living in and cannot really comprehend the rest of theuniverse.
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