On topological rigidity of Alexandrov 3 -spaces
aa r X i v : . [ m a t h . M G ] D ec ON TOPOLOGICAL RIGIDITY OF ALEXANDROV -SPACES NO ´E B ´ARCENAS † AND JES ´US N ´U ˜NEZ-ZIMBR ´ON ‡ A BSTRACT . In this note we prove the Borel Conjecture for closed, irreducibleand sufficiently collapsed three-dimensional Alexandrov spaces. We also poseseveral questions related to the characterization of fundamental groups of three-dimensional Alexandrov spaces, finite groups acting on them and rigidity results.
1. I
NTRODUCTION AND RESULTS
Alexandrov spaces are inner metric spaces which admit a lower sectional cur-vature bound in a synthetic sense. They constitute a generalization of the class ofcomplete Riemannian manifolds with a lower sectional curvature bound and sincetheir introduction they have proven to be a natural setting to address geometric-topological questions of a global nature. Therefore, a central problem is to de-termine whether what is already known in the smooth or topological settings stillholds in Alexandrov geometry.Regarding topological rigidity of spaces, an important conjecture originally for-mulated in the topological manifold category, is the
Borel conjecture . It assertsthat if two closed, aspherical n -manifolds are homotopy equivalent, then they arehomeomorphic. The proof of this conjecture in the three-dimensional case is aconsequence of Perelman’s resolution of the Geometrization Conjecture [22].On the other hand, in high dimension (meaning greater or equal than five), theBorel Conjecture for an aspherical manifold with fundamental group G is con-sequence of the Farrell-Jones Conjecture in Algebraic K - and L -Theory for thegroup G [14]. A lot of effort in geometric topology and surgery theory has beendevoted to prove the Borel conjecture in many cases by these methods, which relyon transversality arguments which are not available for the study of topologicalrigidity of low dimensional manifolds.We will present in the following note a series of questions related to general-izations of the Borel conjecture outside of the manifold category. Steps in thisdirection have been obtained, for example, for CAT ( ) -spaces as a consequenceof the Farrell Jones-Conjecture [1], and in another direction for certain classes oftopological orbifolds [24] as a consequence of classification efforts in three dimen-sional geometric topology beyond manifolds. Date : December 27, 2018.2010
Mathematics Subject Classification.
Key words and phrases.
Collapse, Alexandrov space, Borel conjecture. † Supported by CONACYT- foundational Research Grant 250747, PAPIIT Grant IA 100117. ‡ Supported by a DGAPA-UNAM postdoctoral fellowship.
Negative results concerning the topological rigidity of singular spaces of geo-metric nature, such as the Coxeter complex have been obtained in [23].It is therefore natural to inquire whether the Borel conjecture is still valid forclosed Alexandrov -spaces (cf. [6, Remark 3.12]). These spaces are either topo-logical -manifolds or are homeomorphic to quotients of smooth -manifolds bysmooth orientation-reversing involutions with only isolated fixed points (see [7]).In this article we address the validity of the Borel Conjecture for the class of sufficiently collapsed and irreducible closed Alexandrov -spaces. The definitionof irreducibility for a closed Alexandrov -space was introduced in [9]. Let usrecall that a closed Alexandrov -space X is irreducible if every embedded -sphere in X bounds a -ball and, in the case that the set of topologically singularpoints of X is non-empty, it is further required that every -sided R P bounda K ( R P ) , a cone over a real projective plane R P . The condition related tocollapse is described more precisely by considering the class of spaces A ( D, ε ) ,defined as the class of closed Alexandrov -spaces with curv ≥ − , satisfying that diam ≤ D and vol < ε for given D, ε > . We say that a closed Alexandrov -space X is sufficiently collapsed if X ∈ A ( D, ε ) for a sufficiently small ε with respectto D . Our main result is the following. Theorem A.
For any D > , there exists ε = ε ( D ) > such that, if X, Y ∈ A ( D, ε ) are aspherical and irreducible, then the Borel Conjecture holds for X and Y , that is, if X is homotopy equivalent to Y then X is homeomorphic to Y . We point out that a related result was obtained in [18, Theorem 6.1] where thesecond named author proved the Borel Conjecture for closed Alexandrov -spacesadmitting an isometric circle action. The proof of Theorem A is based on twopoints: the Borel conjecture in the -manifold case and the following result. Theorem B.
For any D > , there exists ε = ε ( D ) > such that, if X ∈ A ( D, ε ) is irreducible and aspherical, then X is homeomorphic to a -manifold. The classification of closed collapsing Alexandrov -spaces due to Mitsuishi-Yamaguchi is a key tool in the proof of Theorem B. The classification of closedAlexandrov -spaces admitting isometric (local) circle actions [10], [18] obtainedby Galaz-Garc´ıa and the second named author also plays a role. The strategy ofproof resembles that of [9, Theorem A]. In fact, without assuming that the spacesin question are sufficiently collapsed or irreducible, the analysis of Section 3.2implies the following result. Corollary C.
Let X be a closed, non-orientable Alexandrov -space with fun-damental group Z , Z ⋊ Z or Z ⊕ Z . Then π ( X ) ≠ . In particular, X is notaspherical. For arbitrary dimension we observe the following result, which is an immediateconsequence of a result of Mitsuishi (see [16, Corollary 5.7] and Theorem 3.1below) and Theorem 3.2 (stated below).
Corollary D.
Let X be a closed, aspherical Alexandrov n -space such that its uni-versal cover is compact. Then X is homeomorphic to a closed n -manifold. N TOPOLOGICAL RIGIDITY OF ALEXANDROV -SPACES 3 In light of Theorem B and Corollary D we propose the following natural con-jecture (cf. Remark 3.4).
Conjecture E.
Every closed, aspherical Alexandrov -space is a -manifold The organization of the article is the following. In Section 2, we briefly recallthe basic structure of Alexandrov -spaces following Galaz-Garc´ıa-Guijarro [7]and the classification of collapsing Alexandrov -spaces of Mitsuishi-Yamaguchi[17]. In Section 3, we prove Theorem B which yields as a consequence the validityof Theorem A. In section 4, we state some questions related to the fundamentalgroups and groups which can act on Alexandrov -spaces, in analogy with similarresults obtained in connection with the topological rigidity of aspherical manifolds. Acknowledgements.
The authors warmly thank Fernando Galaz-Garc´ıa and LuisGuijarro for very helpful conversations. JNZ also thanks Bernardo Villarreal and´Angel Zaldivar for useful communications.2. P
RELIMINARIES
We will assume that the reader is familiar with the general theory of Alexandrovspaces of curvature bounded below and refer to [3] for a more detailed introduction.In this section we will briefly recall some results concerning the structure of closedAlexandrov -spaces. All spaces are considered to be connected throughout thearticle.2.1. Alexandrov -spaces. Let X denote a closed Alexandrov -space and foreach x ∈ X , let Σ x X be the space of directions at x . The space Σ x X is a closedAlexandrov -space with curv Σ x X ≥ (see [3, Theorem 10.8.6]). This implies,via the Bonnet-Myers Theorem (see [3, Theorem 10.4.1]) and the classification ofclosed surfaces that the homeomorphism type of Σ x X is that of a sphere S or thatof a real projective plane R P . A point x ∈ X such that Σ x X is homeomorphic to S is called topologically regular , while a point such that Σ x X is homeomorphic to R P is called topologically singular . We let S ( X ) be the subset of X consisting oftopologically singular points. Then X ∖ S ( X ) is open and dense in X (see [3, The-orem 10.8.5]). Furthermore, the Conical Neighborhood Theorem of Perelman [21]states that each point x ∈ X has a neighborhood which is pointed-homeomorphicto the cone over Σ x X . As a consequence, S ( X ) is a finite set.Topologically, a closed Alexandrov -space X can be described as a compact -manifold M having a finite number of R P -boundary components with a coneover R P attached on each boundary component. In the case that S ( X ) ≠ ∅ thereis an alternative topological description of X as quotient of a closed, orientable,topological -manifold ˜ X by an orientation-reversing involution ι ∶ ˜ X → ˜ X havingonly isolated fixed points. The -manifold ˜ X is called the branched orientabledouble cover of X (see [7, Lemma 1.7]). It is possible to lift the Alexandrov metricon X to an Alexandrov metric on ˜ X having the same lower curvature bound in sucha way that ι is an isometry. In particular, ι is equivalent to a smooth involution on ˜ X regarded as a smooth -manifold (a detailed description of this construction canbe found in [7, Lemma 1.8], [4, Section 2.2] and [11, Section 5]). N. B ´ARCENAS AND J. N ´U ˜NEZ-ZIMBR ´ON
Collapsing Aleandrov -spaces. Let { X i } ∞ i = be a sequence of closed Alexan-drov -spaces with diameters uniformly bounded above by D > and curv X i ≥ k for some k ∈ R . Gromov’s Precompactness Theorem implies that (possibly af-ter passing to a subsequence), there exists an Alexandrov space Y with diameterbounded above by D and curv Y ≥ k such that X i GH Ð→ Y . In the case in which dim Y < , the sequence X i is said to collapse to Y . Similarly, a closed Alexan-drov -space X is a collapsing Alexandrov -space if there exists a sequence ofAlexandrov metrics d i on X such that the sequence {( X, d i )} ∞ i = is a collapsingsequence.The topological classification of closed collapsing Alexandrov -spaces was ob-tained by Mitsuishi-Yamaguchi in [17]. We now give a brief summary of the clas-sification. We denote the boundary of an Alexandrov space Y by ∂Y .In the case in which dim Y = (cf. [17, Theorems 1.3, 1.5]), for sufficientlybig i , X i is homeomorphic to a generalized Seifert fibered space Seif ( Y ) (see [17,Definition 2.48]). In the case in which ∂Y ≠ ∅ , Seif ( Y ) is attached with a finitenumber of generalized solid tori and Klein bottles (see [17, Definition 1.4]).In the event that dim Y = and ∂Y = ∅ (cf. [17, Theorem 1.7]), for big enough i , X i is homeomorphic to an F -fiber bundle over S , where F is homeomorphicto one of the spaces T , K , S or R P . On the other hand, if ∂Y ≠ ∅ (cf. [17,Theorem 1.8]), X i is homeomorphic to a union of two spaces B and B ′ with oneboundary component, glued along their homeomorphic boundaries, where ∂B isone of the spaces T , K , S or R P . The pieces B and B ′ are determined asfollows:(i) If ∂B ≅ S then B and B ′ are homeomorphic to one of: a -ball D , a -dimensional projective space with the interior of a -ball removed R P ∖ int D or B ( S ) , a space homeomorphic to a small metric ball of an S -soul of an open non-negatively curved Alexandrov space L ( S ; 2 ) (cf. [17,Corollary 2.56]).(ii) If ∂B ≅ R P then B and B ′ are homeomorphic to a closed cone over aprojective plane K ( R P ) .(iii) If ∂B ≅ T then B and B ′ are homeomorphic to one of S × D , S × Mo ,the orientable non-trivial I -bundle over K , K ˜ × I or B ( S ) , a spacehomeomorphic to a small metric ball of an S -soul of an open non-negativelycurved Alexandrov space L ( S ; 4 ) (cf. [17, Corollary 2.56]).(iv) If ∂B ≅ K then B and B ′ are homeomorphic to one of S ˜ × D thenon-orientable D -bundle over S , K ˆ × I the non-orientable non-trivial I -bundle over K , the space B ( pt ) defined in [17, Example 1.2], or B ( R P ) ,a space homeomorphic to a small metric ball of an R P -soul of an opennon-negatively curved Alexandrov space L ( R P ; 2 ) (cf. [17, Corollary2.56]).Finally, if dim Y = (cf. [17, Theorem 1.9]), then for i sufficiently big, X i ishomeomorphic to either a generalized Seifert fiber space Seif ( Z ) , (where Z is a -dimensional Alexandrov space with curv Z ≥ ), a space appearing in the cases N TOPOLOGICAL RIGIDITY OF ALEXANDROV -SPACES 5 in which dim Y = , or a non-negatively curved Alexandrov space with finitefundamental group.In order to provide information on the homotopy groups of some of the spacesappearing in the previous classification we now recall the celebrated Soul Theoremfor Alexandrov spaces due to Perelman [20, §6] Theorem 2.1 (Soul Theorem) . Let X be a compact Alexandrov space of curv ≥ with ∂X ≠ ∅ . Then there exists a totally convex, compact subset S ⊂ X , called thesoul of X with ∂S = ∅ which is a strong deformation retract of X . The spaces B ( S ) , B ( S ) , B ( pt ) and B ( R P ) admit Alexandrov metrics of curv ≥ . Therefore the Soul Theorem can be applied. Moreover, given that thesoul is a strong deformation retract of the space, in particular we have a homo-topy equivalence. Whence, π k ( B ( S )) ≅ π k ( S ) , π k ( B ( S )) ≅ π k ( S ) , while π k ( B ( pt )) = and π k ( B ( R P )) ≅ π k ( R P ) for all k ≥ .3. P ROOFS
We proceed to prove Theorem B. As stated in the Introduction, this result readilyimplies our main result, Theorem A.
Proof of Theorem B.
We will proceed by contradiction. Let us suppose that theresult in question does not hold. Then, there exists a sequence of closed, irre-ducible and aspherical Alexandrov -spaces { X i } ∞ i = of curv X i ≥ − , satisfyingthat diam X i ≤ D and vol X i i →∞ Ð→ which are not homeomorphic to -manifolds.Therefore by Gromov’s precompactness Theorem we can assume (possibly pass-ing to a non-relabeled subsequence) that X i collapses in the Gromov-Hausdorfftopology to a closed Alexandrov space Y of dimension < . We will split the proofin three cases depending on whether dim Y = , , and obtain a contradiction ineach case. Observe that by our contradiction assumption in the following analysiswe will exclude any Alexandrov -spaces appearing in the classification [17] thatare homeomorphic to -manifolds.3.1. -dimensional limit space. In the case that dim Y = , we need to addresstwo further cases depending on whether X i contains singular fibers with neighbor-hoods of type B ( pt ) or not. In the case in which X i does not contain fibers oftype B ( pt ) , then by [10, Corollary 6.2], the collapse X i GH Ð→ Y is equivalent to theone obtained by collapsing along the orbits of a local circle action on X i . There-fore, by [10, Theorem B] X i is homeomorphic to a connected sum of the form M ( R P ) ⋯ ( R P ) , where M is a closed -manifold admitting alocal circle action. It now follows from [9, Lemma 5.1] that either X i is homeo-morphic to M or to Susp ( R P ) . Since we are assuming X i is not homeomorphicto a -manifold, we conclude that X i is homeomorphic to Susp ( R P ) . However, Susp ( R P ) is not aspherical, as a combination of the suspension isomorphism andthe Hurewicz Theorem yields that π ( Susp ( R P )) is isomorphic to Z .We move on to the case in which X i contains fibers with tubular neighborhoodsof the form B ( pt ) . Let us consider the branched orientable double cover ˜ X i of X i . N. B ´ARCENAS AND J. N ´U ˜NEZ-ZIMBR ´ON
We now recall that, by [9, Corollary 3.2] X i collapses if and only if the sequenceof branched orientable double covers ˜ X i collapses. Then, by [10, Corollary 6.2],the connected sum decomposition in [10, Theorem B], and taking into account theorientability of ˜ X i we have that ˜ X i is homeomorphic to a connected sum of theform(3.1) ( ϕ S × S ) ( n j = L ( α j , β j )) where L ( α j , β j ) denotes a lens space determined by the Seifert invariants ( α j , β j ) (see [19, Section 1.7]).It was proved in [9, Page 14] that, in this situation, the connected sum (3.1)cannot contain both lens spaces and copies of S × S , that is, either X i is home-omorphic to a connected sum of lens spaces or to a connected sum of copies of S × S . However, the irreducibility assumption implies, as in [9, Case 5.6], thatthe expression (3.1) cannot contain nj = L ( α j , β j ) as a connected summand andtherefore only copies of S × S appear in the connected sum 3.1. Therefore X i is homeomorphic to ϕ S × S . Furthemore, it follows from the irreducibility of X i as in [9, Case 5.7] that ˜ X i is homeomorphic to S × S . Hence, it suffices toconsider the case in which X i is a quotient of S × S by an orientation reversinginvolution ι ∶ S × S → S × S having only isolated fixed points.Let us consider S × S as a subspace of R × C and denote its points by (( x, y, z ) , w ) . The classification [25] of involutions on S × S yields that the in-volution ι i on ˜ X i satisfying that ˜ X i / ι i ≅ X i is equivalent to the involution definedby (( x, y, z ) , w ) ↦ ((− x, − y, z ) , w ) which has four fixed points. The quotient space of this involution is homeomorphicto Susp ( R P ) ( R P ) (see incise (a) of Case (2) in Page 7 of [7]). Then itfollows, as in the proof of [18, Theorem 6.1], that X i is not aspherical.3.2. -dimensional limit space. In this case, for sufficiently big i , X i is homeo-morphic to a gluing of two pieces B and B ′ appearing in the classification [17],along their isometric boundaries. As X i is not homeomorphic to a manifold,at least one of the following pieces must appear in the decomposition: B ( S ) , K ( R P ) , B ( S ) , B ( pt ) and B ( R P ) . We will show in this section that everypossible space X i having one of these pieces cannot be aspherical. Case 3.1 (One of the pieces is B ( S ) ) . The possible pieces in this case are D , R P ∖ int D and B ( S ) . By [17, Remark 2.62], B ( S ) is homeomorphic to Susp ( R P )∖ D . Whence the possible spaces arising by gluing with this piece arehomeomorphic to Susp ( R P ) , R P ( R P ) and Susp ( R P ) ( R P ) .All of these spaces have Susp ( R P ) as a connected summand, and therefore thearguments of [18, Theorem 6.1] show that they are not aspherical, a contradiction. Case 3.2 (One of the pieces is K ( R P ) ) . The only piece is the closed cone K ( R P ) and therefore, the space X i in this case is homeomorphic to Susp ( R P ) . N TOPOLOGICAL RIGIDITY OF ALEXANDROV -SPACES 7 As previously mentioned, this space is not aspherical and then a contradiction isensued.As a first step to deal with the remaining cases, we observe that H ( X i ) ≠ whenever X i contains a piece of the form B ( S ) or B ( pt ) a fact that followsfrom the following result due to Mitsuishi (see [16, Corollary 5.7]). The resultis originally stated for the more general class of N B -spaces (see [16, Definition1.6]). For simplicity we restate it here for Alexandrov spaces only.
Theorem 3.1 (Mitsuishi) . Let X be a closed, connected Alexandrov n -space. If X is non-orientable then the torsion subgroup of H n − ( X ; Z ) is isomorphic to Z and, in particular, H n − ( X ; Z ) is non-zero. In order to obtain that π ( X i ) ≠ using the information that H ( X i ) ≠ wewill use the following classical theorem proved by Hopf in [12, Theorem a), Page257]. Theorem 3.2 (Hopf) . Let X be a CW -complex with finitely many cells. Then,there exists an exact sequence (3.2) π ( X ) → H ( X ) → H ( Bπ ( X ) ; Z ) → . Here, Bπ ( X ) denotes a model for the classifying space of the fundamentalgroup π ( X ) , which is characterized up to homotopy by being an aspherical CW -complex having the same fundamental group as X . As π ( X ) depends on thepieces B and B ′ we will split the following analysis to go over every possibility. Case 3.3 (One of the pieces is B ( S ) ) . The possible pieces B and B ′ with X i ≅ B ∪ B ′ in this case are S × D , S × Mo , K ˜ × I and B ( S ) . We assume that B ′ = B ( S ) is fixed. Case 3.3.1 ( B = B ( S ) ) . If both pieces of the decomposition of X i are homeo-morphic to B ( S ) , then Van Kampen’s Theorem readily implies that π ( X i ) = .Moreover, by Theorem 3.1, H ( X i ) ≠ . Therefore, by Hurewicz’s Theorem, π ( X i ) ≠ , which contradicts the asphericity of X i . Case 3.3.2 ( B = S × Mo ) . The fundamental group of X i in this case is Z ⊕ Z ,as calculated from Van Kampen’s Theorem. A model for the classifying space B ( Z ⊕ Z ) is the torus T . Hence, the sequence (3.2) becomes(3.3) π ( X i ) → H ( X i ) → Z → . Here, we have used the fact that H ( T ) ≅ Z (as a consequence of the ori-entability of T ). Therefore, if π ( X i ) = we would obtain that H ( X i ) ≅ Z ,in particular yielding that H ( X i ) is torsion-free. This contradicts Theorem 3.1.Therefore, π ( X i ) ≠ which is a contradiction to the asphericity of X i . Case 3.3.3 ( B = S × D ) . In this case, a computation via the Van Kampen’sTheorem yields that π ( X i ) ≅ Z . Now, a model for B Z is the circle S . Hence,Hopf’s exact sequence (3.2) takes the form(3.4) π ( X i ) → H ( X i ) → → . N. B ´ARCENAS AND J. N ´U ˜NEZ-ZIMBR ´ON
Therefore, the morphism π ( X i ) → H ( X i ) ≠ is surjective, implying that π ( X i ) ≠ . This contradicts the assumption that X i is aspherical. Case 3.3.4 ( B = K ˜ × I ) . As in the previous cases, using Van Kampen’s theorem itfollows that the fundamental group of X i is that of K ˜ × I . Since I is contractible, π ( K ˜ × I ) ≅ π ( K ) ≅ Z ⋊ Z . We now note that a model for B ( Z ⋊ Z ) is theKlein bottle. Therefore by the non-orientability of K , the Hopf’s sequence (3.2)becomes(3.5) π ( X i ) → H ( X i ) → → . Whence, as in the previous case, π ( X i ) ≠ . Case 3.4 (One of the pieces is B ( pt ) ) . In this case the possible pieces taking therole of B and B ′ are S ˜ × D , K ˆ × I , B ( pt ) and B ( R P ) . We will exclude thespace B ( pt ) ∪ B ( R P ) in this case as it will be considered below. Let us observethat by Van Kampen’s Theorem, the possible fundamental groups of X i in this caseare the same that appear in the case that one of the pieces is B ( S ) . Therefore, acontradiction to the asphericity of X i is obtained for B ( pt ) ∪ B ( pt ) as in Case3.3.1, for B ( pt ) K ˆ × I as in Case 3.3.4, and for B ( pt ) ∪ S ˜ × D as in Case (3.3.3).We now address the remaining case in which one of the pieces in the decompo-sition of X i is B ( R P ) . Case 3.5 (One of the pieces is B ( R P ) ) . Under this assumption, the possiblepieces B and B ′ forming X i are S ˜ × D , K ˆ × I , B ( pt ) and B ( R P ) . As somepossibilities overlap with the previous Case (3.4), we only consider the spaces S ˜ × D ∪ B ( R P ) , K ˆ × I ∪ B ( R P ) , B ( pt ) ∪ B ( R P ) and B ( R P ) ∪ B ( R P ) here. To address the question of asphericity of these spaces we apply the followingresult (see [13, Lemma 4.1]) Theorem 3.3.
The fundamental group of an aspherical finite-dimensional CW -complex is torsion-free. An analysis via Van Kampen’s Theorem yields that the possible fundamentalgroups of X i in this case are Z ∗ π ( K ) Z , ( Z ⋊ Z )∗ π ( K ) Z , Z and Z ∗ π ( K ) Z .It is immediate to check that these groups have non-zero torsion, as at least one ofthe factors in the amalgamated products has non-zero torsion. Therefore Theorem3.3 yields that X i cannot be aspherial, a contradiction.3.3. -dimensional limit space. In this case, if X i is a generalized Seifert fiberedspace then the contradiction is obtained as in Section 3.1. If X i is homeomorphic toa space appearing in the -dimensional limit case, then the contradiction is obtainedas in Section 3.2. The remaining cases are non-negatively curved (non-manifold)Alexandrov -spaces with finite fundamental group. In these cases, if π ( X i ) = ,Theorem 3.1, implies that X i is not aspherical. Furthermore, if π ( X i ) is non-trivial then 3.3 yields that X i is not aspherical. Hence, in every case we obtain acontradiction to asphericity and the result is settled. (cid:3) Remark 3.4.
In light of Theorem B, a natural conjecture would be that a closed,aspherical Alexandrov -space X is homeomorphic to a -manifold. This is indeed N TOPOLOGICAL RIGIDITY OF ALEXANDROV -SPACES 9 the case whenever X is simply-connected as a consequence of incise (2) of [16,Corollary 5.7] and the Hurewicz Theorem.4. Q UESTIONS ON FUNDAMENTAL GROUPS AND ACTIONS ON A LEXANDROV
PACES
A lot of effort has been devoted in geometric topology to the development ofcharacterization of fundamental groups of manifolds and spaces which are topo-logically rigid. On the other hand, a similar fruitful effort has been devoted to thecharacterization of groups which act on geometrically defined classes of manifolds.We will now consider a series of questions inspired by the study of topologicalrigidity of manifolds and their extrapolation to (possibly singular) Alexandrov -spaces.These questions evolve from topological or geometric rigidity results such asthe Borel Conjecture for the fundamental group G of an aspherical manifold, intothe characterization of fundamental groups of such spaces through concepts ofgeometric group theory or group cohomology.Specifically related to group cohomology, it is an open conjecture originallyposed by Wall [26] that every poincar´e duality group of dimension is the funda-mental group of a three dimensional manifold.See [5], [8] for a modern discussions on the subject. Here, we present the fol-lowing question: Question 4.1.
Let G be a Poincar´e duality group which is the fundamental groupof an aspherical Alexandrov -space. Is it the fundamental group of an orientablethree dimensional manifold?Question 4.1 follows readily from Conjecture E.In the direction of characterizations of groups acting on manifolds, it is a classi-cal result by Wall [26], that the finite groups acting on three-dimensional Poincar´ecomplexes have periodic cohomology of period . It is known that the symmetricgroup on three letters Σ cannot be realized by any honest manifold [15]. Question 4.2.
Which finite groups act by homeomorphisms on Alexandrov -spaces?The study of geometric and large scale geometric properties of fundamentalgroups of three dimensional manifolds in connection with topological rigidity hasbeen oriented in recent times to the characterization of the map involved in thehomotopy equivalence referred to in the statement. Consider as an example thefollowing question. Question 4.3.
Let f ∶ M → N be a map between three dimensional, asphericalmanifolds with boundary inducing an epimorphism in fundamental groups. Underwhich conditions is f homotopic to a homeomorphism.?This problem has been studied using simplicial volume, specifically degree the-orems, and more recently, Agol’s solution to the virtually fibering conjecture byBoileau and Friedl [2]. (The epimorphic condition comes from the fact that a degree one map inducessuch an epimorphism due to Poincar´e Duality and the loop theorem).4.1. Consequences of Agol’s virtually fibering Theorem.
The following theo-rem was proved by Boileau and Friedl, [2].
Theorem F.
Let f ∶ M → N be a proper map between aspherical manifolds witheither toroidal or empty boundary. Assume that N is not a closed graph manifold,and that f induces an epimorphism on the fundamental group.Then f is homotopicto a homeomorphism if any of the following two conditions are met: ● For each H finite index, subnormal subgroup of π ( N ) , the ranks of H and f − ( H ) agree. ● For each finite cover ̃ N of N , the Heegard genus of ̃ N and ̃ M agree. Question 4.4.
Let f ∶ M → N be an oriented map between oriented, asphericalAlexandrov spaces. What are the conditions on f , M , and N for f be homotopicto homemomorphism? R EFERENCES
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