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On the Finiteness of the Classifying Space of Diffeomorphisms Of ON THE FINITENESS OF THE CLASSIFYING SPACE OF DIFFEOMORPHISMS OF REDUCIBLE THREE MANIFOLDS Sam Nariman Abstract In this paper, we use homological techniques and a theorem of Thurston to study homological finiteness of BDiffpM, rel Bq when M is a reducible 3-manifold with a non-empty boundary that has distinct irreducible factors. Kontsevich ([Kir95, Problem 3.48]) conjectured that BDiffpM, rel Bq has the homotopy type of a fi- nite CW complex for all 3-manifolds with non-empty boundary. Hatcher-McCullough ([HM97]) proved this conjecture when M is irreducible. We show that when M is reducible with non-empty boundary that has distinct irreducible factors, BDiffpM, rel Bq is homology isomorphic to a finite CW-complex. 1. Introduction For a closed surface Σg of genus g ą 1, it is well-known that the classifying space BDiffpΣg q is rationally equivalent to Mg , the moduli space of Riemann surfaces of genus g. Therefore, in particular the rational homology groups of BDiffpΣg q vanish above a certain degree and in fact more precisely they vanish above 4g ´5, which is the virtual cohomological dimension of the mapping class group ModpΣg q. And for a surface Σg,k with k ą 0 boundary components, BDiffpΣg,k , rel Bq is in fact homotopy equivalent to Mg,k . Therefore, BDiffpΣg,k , rel Bq has a arXiv:2104.12338v1 [math.GT] 26 Apr 2021 finite dimensional model. For 3-manifolds, one cannot expect a similar statement to hold for closed 3-manifolds but Kontsevich ([Kir95, Problem 3.48]) conjectured for 3-manifolds with non-empty boundary, the classifying space BDiffpM, rel Bq has a finite dimensional model. This conjecture is known to hold for irreducible 3-manifolds with non-empty boundary ([HM87]). We shall prove a homological finiteness of these classifying spaces for certain reducible 3-manifolds. Theorem 1.1. Let M be a reducible 3-manifolds with non-empty boundary that has distinct irreducible factors. Then the classifying space BDiffpM, rel Bq is homology isomorphic to a finite CW-complex. 1 2 SAM NARIMAN In the irreducible case, in fact, the homotopy type of the group DiffpMq is very well studied. When M admits one of the Thurston’s geometries, there has been an encompassing program known as the generalized Smale’s conjecture that relates the homotopy type of DiffpMq with the isometry group of the corresponding geometry (for more details and history, see the discussions in Problem 3.47 in [Kir95] and Sections 1.2 and 1.3 in [HKMR12]). For S3, it was proved by Hatcher ([Hat83]) and for Haken 3-manifolds, it is a consequence of Hatcher’s work and also understanding the space of incompressible surfaces ([Wal68, Hat76, Iva76]) inside such manifolds. Recently Bamler and Kleiner ([BK19]) used Ricci flow techniques to settle the generalized Smale’s conjecture for all 3-manifolds admitting the spherical geometry. The only case that has not yet appeared in print is the case of Nil manifolds. Bamler and Kleiner mentioned in their paper ([BK19]) that they have a work in progress ([BK]) that also addresses the Nil case. Hence, this recent body of work using Ricci flow techniques addresses all cases of the generalized Smale’s conjecture. Recall that a 3-manifold M is called prime if it is not homeomorphic to a connected sum of more than one 3-manifolds so that none of which is homeomorphic to the 3-sphere. The prime decomposition theorem says that every closed 3-manifold is homeomorphic to connected sum of prime manifolds. A prime closed 3-manifold is either homeomorphic to S1 ˆ S2 or it is irreducible (i.e. every embedding S2 bounds a ball). On other hand, the geometric manifolds are the building blocks for irre- ducible manifolds. Given the generalized Smale’s conjecture, we have a good understanding of the homotopy type of the diffeomorphism groups for these atomic pieces. On the other hand, the JSJ and geometric de- composition theorems give a way to cut an irreducible manifold along embedded tori into these building blocks. If the JSJ decomposition is non-trivial for an irreducible manifold, then it will be Haken whose dif- feomorphism groups are well studied. Hence, given that we also know the homotopy type of the diffeomorphism group of S1 ˆS2 by Hatcher’s theorem ([Hat81]), we have a good understanding of the homotopy type of diffeomorphism group of prime manifolds. In the reducible case, the prime decomposition theorem cuts the manifold along separating spheres into its prime factors. The difficulty, however, in understanding the reducible case is to related the diffeomorphism group of a reducible manifold to the diffeomorphisms of its prime factors. Idea of the proof. C´esar de S´aand Rourke ([CdSR79]) made a pro- posal to describe the homotopy type of DiffpMq in terms of the homo- topy type of diffeomorphisms of the prime factors and an extra factor of the loop space on “the space of prime decompositions”. Hendriks- Laudenbach ([HL84]) and Hendriks-McCullough ([HM87]) found a 3 model for this extra factor. Later Hatcher, in an interesting unpub- lished note, proposed a finite dimensional model for this “space of prime decompositions” and more interestingly, he proposed that there should be a “wrong-way map” between BDiffpMq and the classifying space of diffeomorphisms of prime factors. His approach, if completed, would also solve Kontsevich’s question for reducible 3-manifolds such that all the irreducible factors have non-empty boundary. Instead of trying to build this wrong way map, we shall find a model of the classifying space given by the homotopy colimit over a finite diagram of finite CW com- plexes inspired by the techniques that Kathryn Mann and the author ([MN20]) used to study the second homology of BDiffpMq. For technical simplicity, we work with the homeomorphism groups instead of diffeomorphism groups. The reason is that Cerf ([Cer61]) assumed Smale’s conjecture which was later proved by Hatcher ([Hat83]) to show that in these low dimensions, the inclusion DiffpMq ãÑ HomeopMq is in fact a weak homotopy equivalence. On the other hand, in all dimensions, by Mather-Thurston’s theorem ([Thu74, Corollary (b) of theorem 5]) for homeomorphisms, we have the natural map BHomeoδpMqÑ BHomeopMq, which is an acyclic map and in particular it induces a homology isomor- phism in all degrees where HomeoδpMq is the same group of homeomor- phisms of M equipped with the discrete topology. The same statement also holds in the relative case. Hence to prove the main theorem, we use a homological approach where we consider the action of HomeoδpM, rel Bq on a semisimplicial set given by the complex of parametrized separating spheres, to give a model for BHomeoδpM, rel Bq suitable for an inductive argument to prove the main theorem. Theorem 1.2. There exists a finite ∆-complex Ond pMq (see Defi- nition 2.4) whose vertices are isotopy classes of separating spheres and its simplices can be represented by disjoint separating spheres in differ- ent isotopy classes. For a representative of a simplex σ in Ond pMq, let HomeopM, σq be those homeomorphisms that preserve each sphere in the simplex σ setwise. Then we have BHomeopMq is homology isomorphic to hocolim pBHomeopM, σqq, σPOnd pMq and similarly BHomeopM, rel Bq is homology isomorphic to hocolimσPOnd pMqpBHomeopM, σ, rel Bqq. As we shall see Ond pMq is a finite ∆-complex. Hence to prove The- orem 1.1 we use what is known about the irreducible case and also the generalized Smale’s conjecture, to show that for each simplex σ, the 4 SAM NARIMAN classifying space BHomeopM, σq has the homotopy type of a finite CW complex. Then in particular the spectral sequence for homotopy colimit implies that the homology of BHomeopM, rel Bq vanishes in high enough degrees. Acknowledgment. The author was partially supported by NSF grants DMS-1810644 and DMS-2113828. He thanks Sander Kupers for his comments on the first draft of this paper. 2. Separating sphere complexes Let M be a reducible 3-manifold with possibly non-empty boundary in this section. We want to study the group homology of HomeoδpM, rel Bq via its action on a contractible simplicial complex given by parametrized separating spheres in M. This action helps us to determine how the classifying space of the prime factors of M should glue together to give BHomeoδpM, rel Bq as in Theorem 1.2. Definition 2.1. Let SpMq be a simplicial complex whose vertices are given by locally flat embeddings φ: S2 ãÑ M whose images are separating spheres, and simplices are given by collection of embeddings whose images are disjoint. Note that in a simplex there could be vertices that are given by isotopic spheres and since they are disjoint they bound an embedded S2 ˆ r0, 1s. Proposition 2.2. The simplicial complex SpMq is contractible. Proof. This is a simpler version of Lemma 3.4 in [MN20]. For the convenience of the reader, we give the argument. It is enough to show that for all k, any continuous map f : Sk Ñ SpMq is nullhomotopic. To do so, we can assume that f is a PL map with respect to a triangulation K of Sk and we shall change f up to a simplicial homotopy such that for a vertex v in SpMq, v becomes a cone over f pSk q in SpMq. It is easy to arrange also in this PL setting to modify the map f via a simplicial homotopy given moving vertices to their parallel copies to make sure that the images of vertices of f pKq are pairwise transverse 1 (see [Nar20, Lemma 3.31 and Lemma 4.3] for treating transversality in the locally flat settings).
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