On the finiteness of the classifying space of diffeomorphisms of reducible three manifolds

For a closed surface $\Sigma_{g}$ of genus $g>1$, it is well-known that the classifying space $\mathrm{BDiff}(\Sigma_{g})$ is rationally equivalent to $\mathcal{M}_{g}$, the moduli space of Riemann surfaces of genus $g$. Therefore, in particular, the rational homology groups of $\mathrm{BDiff}(\Sigma_{g})$ vanish above a certain degree, and in fact, more precisely they vanish above degree $4g-5$, which is the virtual cohomological dimension of the mapping class group $\text{Mod}(\Sigma_{g})$. And for a surface $\Sigma_{g,k}$ with $k>0$ boundary components, the classifying space $\mathrm{BDiff}(\Sigma_{g,k},\text{rel }\partial)$ is in fact homotopy equivalent to the moduli space $\mathcal{M}_{g,k}$. Therefore, $\mathrm{BDiff}(\Sigma_{g,k},\text{rel }\partial)$ has the homotopy type of a finite-dimensional CW-complex.

Similarly, Kontsevich ([Kir95, Problem 3.48]) conjectured for compact $3$-manifold $M$ with non-empty boundary, the classifying space $\mathrm{BDiff}(M,\text{rel }\partial)$ has a finite-dimensional model. This conjecture is known to hold for irreducible $3$-manifolds with non-empty boundary ([HM97]). In this paper, we shall prove the homological finiteness of these classifying spaces for reducible $3$ manifolds with a condition on its boundary.

Throughout this paper, for brevity, we write $\mathrm{Diff}(M,\text{rel }\partial)$ and $\text{{Homeo}}(M,\text{rel }\partial)$ to denote the smooth orientation preserving diffeomorphisms and orientation preserving homeomorphisms respectively whose supports are away from the boundary $\partial M$ and in general when we use $\text{rel }X$ in the diffeomorphism group, for some $X\subset M$, we mean those diffeomorphisms or homeomorphisms whose supports are away from $X$.

In the irreducible case, the homotopy type of the group $\mathrm{Diff}(M)$ is very well studied. When $M$ admits one of Thurston’s geometries, there has been an encompassing program known as the generalized Smale’s conjecture that relates the homotopy type of $\mathrm{Diff}(M)$ 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 $\mathbb{S}^{3}$, 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, BK21]) used Ricci flow techniques to settle the generalized Smale’s conjecture for all $3$-manifolds admitting the spherical geometry and in the Nil geometry. 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 diffeomorphic to a connected sum of more than one $3$-manifold so that none of which is diffeomorphic to the $3$-sphere. The prime decomposition theorem says that every closed $3$ manifold is diffeomorphic to the connected sum of prime manifolds. A prime closed $3$-manifold is either diffeomorphic to $\mathbb{S}^{1}\times\mathbb{S}^{2}$ or it is irreducible (i.e. every embedding $\mathbb{S}^{2}$ bounds a ball). On the other hand, geometric manifolds are the building blocks for irreducible manifolds. Given the generalized Smale’s conjecture, we have a good understanding of the homotopy type of the diffeomorphism groups for these atomic pieces. And the JSJ and geometric decomposition theorems (see [Neu96, Chapter 2, section 6] for the statement of these 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 diffeomorphism groups are well studied. Hence, given that we also know the homotopy type of the diffeomorphism group of $\mathbb{S}^{1}\times\mathbb{S}^{2}$ by Hatcher’s theorem ([Hat81]), we have a complete 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 relate the diffeomorphism group of a reducible manifold to the diffeomorphisms of its prime factors.

César de Sá and Rourke ([CdSR79]) proposed to describe the homotopy type of $\mathrm{Diff}(M)$ in terms of the homotopy 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 model for this extra factor. Later Hatcher, in an interesting unpublished 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 $\mathrm{BDiff}(M)$ and the classifying space of diffeomorphisms of prime factors.

Hatcher’s approach, if completed, would also solve Kontsevich’s conjecture in the special case of reducible $3$ manifolds such that all the irreducible factors have non-empty boundaries. So our result is the homological version of what Hatcher intended to prove about Kontsevich’s conjecture. However, instead of trying to build this wrong way map, we take the geometric group theory approach by letting the abstract group of diffeomorphisms act on a “huge" simplicial complex inspired by the techniques that Kathryn Mann and the author ([MN20]) used to study the second homology of $\mathrm{BDiff}(M)$.

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 $\mathrm{Diff}(M)\hookrightarrow\mathrm{Homeo}(M)$ 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

(1.1)

$$\mathrm{B}\text{{Homeo}}^{\delta}(M)\to\mathrm{B}\text{{Homeo}}(M),$$

which is an acyclic map and in particular it induces a homology isomorphism in all degrees. The same statement also holds in the relative case in particular relative to the boundary when it is non-empty (see [McD80]).

Hence to prove the main theorem, we use a homological approach where we consider the action of $\text{{Homeo}}^{\delta}(M,\text{rel }\partial)$ on a simplicial complex $\mathcal{S}(M)$ given by the complex of essential spheres, to give a model for $\mathrm{B}\text{{Homeo}}^{\delta}(M,\text{rel }\partial)$ suitable for an inductive argument to prove the main theorem.

In this section we assume that $M$ is a compact reducible $3$-manifold with a non-empty boundary. We assume that we do not have spherical boundary components in order to have a prime decomposition in the presence of the boundary ([Hem76, Chapter 3]). To induct on the number of its prime factors to study the homological finiteness of $\mathrm{B}\text{{Homeo}}^{\delta}(M,\text{rel }\partial)$, we shall first construct a simplicial complex $\mathcal{S}(M)$ on which $\text{{Homeo}}^{\delta}(M,\text{rel }\partial)$ acts simplicially.

Note that the group $\text{{Homeo}}^{\delta}(M,\text{rel }\partial)$ acts on $\mathcal{S}(M)$ simplicially. We shall prove in 2.6 that $\mathcal{S}(M)$ is contractible. Therefore, the homotopy quotient $\mathcal{S}(M)/\!\!/\text{{Homeo}}^{\delta}(M,\text{rel }\partial)$ is homotopy equivalent to $\mathrm{B}\text{{Homeo}}^{\delta}(M,\text{rel }\partial)$. The stabilizer of each simplex in $\mathcal{S}(M)$ is the subgroup of $\text{{Homeo}}^{\delta}(M,\text{rel }\partial)$ that fixes a set of essential spheres pointwise so it is isomorphic to diffeomorphism group of a $3$-manifold whose connected components have fewer prime factors. But one issue is that $\mathcal{S}(M)$ has simplices of arbitrary large dimensions since we allow parallel spheres. To account for this infinite dimensionality, we use the simplicial complex that Hatcher and McCullough defined in [HM90, Section 1].

The mapping class group $\text{Mod}(M,\text{rel }\partial)=\pi_{0}(\text{{Homeo}}(M,\text{rel }\partial))$ acts on $\mathcal{S}_{nd}(M)$ simplicially. The complex $\mathcal{S}_{nd}(M)$ is finite-dimensional and also by Hatcher-McCullough’s theorem ([HM90, Proposition 2.2]) the set of the orbits of the action of $\text{Mod}(M,\text{rel }\partial)$ on simplices is also finite.

To briefly recall why this is the case, they use a theorem of Scharlemann (see [Bon83, Appendix A, Lemma A.1]) to find a “normal" representative of each orbit. Let the prime decomposition of $M$ be given by $P_{1}\#P_{2}\#\cdots\#P_{r}\#^{g}\mathbb{S}^{1}\times\mathbb{S}^{2}$ where $g$ summands are diffeomorphic to $\mathbb{S}^{1}\times\mathbb{S}^{2}$. Let $B$ be a punctured $3$-cell having ordered $r+2g$ boundary components so that $M$ is obtained by gluing $P_{i}\backslash\text{int}(\mathbb{D}^{3})$ to $i$-th sphere boundaries for $1\leq i\leq r$ and $g$ copies of $\mathbb{S}^{2}\times[0,1]$ are glues along the remaining $2g$ boundary components (see [Bon83, Appendix A, Lemma A.1] for more detail).

Now as Hatcher and McCullough observed in [HM90, Proposition 2.2], there are finitely many isotopy classes of essential spheres in $B$ since they are determined by the way they partition the boundary components of $B$. And this observation implies the finiteness of the orbits of the action of $\text{Mod}(M,\text{rel }\partial)$ on simplices of $\mathcal{S}_{nd}(M)$.

The skeletal filtration on $\mathcal{S}_{nd}(M)$ induces a filtration on the quotient space

(2.1)

$$\mathcal{F}_{0}\subset\mathcal{F}_{1}\subset\dots\subset\mathcal{F}_{n}=\mathcal{S}_{nd}(M)/\text{Mod}(M,\text{rel }\partial),$$

and by Hatcher and McCullough’s observation, the filtration quotients are given by the wedge of a finite number of spheres. Let $O_{p}$ be the set of orbits of the action of $\text{Mod}(M,\text{rel }\partial)$ on $p$-simplices of $\mathcal{S}_{nd}(M)$. So $\mathcal{F}_{p}-\mathcal{F}_{p-1}$ is homeomorphic to the disjoint union of open $p$-simplices indexed by $O_{p}$.

Now there is a natural simplicial map $\mathcal{S}(M)\to\mathcal{S}_{nd}(M)$ which is equivariant with respect to the map $\text{{Homeo}}^{\delta}(M,\text{rel }\partial)\to\text{Mod}(M,\text{rel }\partial)$. So we have a map $\mathcal{S}(M)/\text{{Homeo}}^{\delta}(M,\text{rel }\partial)\to\mathcal{S}_{nd}(M)/\text{Mod}(M,\text{rel }\partial)$ which in turn induces a map

$$\eta\colon\mathcal{S}(M)/\!\!/\text{{Homeo}}^{\delta}(M,\text{rel }\partial)\to\mathcal{S}_{nd}(M)/\text{Mod}(M,\text{rel }\partial).$$

The preimage filtration $\mathcal{G}_{p}=\eta^{-1}(\mathcal{F}_{i})$ on $\mathcal{S}(M)/\!\!/\text{{Homeo}}^{\delta}(M,\text{rel }\partial)$ induces a spectral sequence

$$E^{1}_{p,q}=H_{p+q}(\mathcal{G}_{p}/\mathcal{G}_{p-1};\mathbb{Z})\Longrightarrow H_{p+q}(\mathrm{B}\text{{Homeo}}^{\delta}(M,\text{rel }\partial);\mathbb{Z}).$$

To prove the classifying space $\mathrm{B}\text{{Homeo}}^{\delta}(M,\text{rel }\partial)$ has finitely many nonzero homology groups which are each finitely generated, it is enough to prove the same for the filtration quotients $\mathcal{G}_{p}/\mathcal{G}_{p-1}$. And this is equivalent to the following theorem.

The filtration $\mathcal{G}_{i}$’s are sub-CW-complexes of realization of the semisimplicial set given by the bar construction that realizes to the homotopy quotient $\mathcal{S}(M)/\!\!/\text{{Homeo}}^{\delta}(M,\text{rel }\partial)$. Hence, the inclusions $\mathcal{G}_{p-1}\hookrightarrow\mathcal{G}_{p}$ are cofibrations. On the other hand, given 2.5, the homotopy type of $\eta^{-1}(x)$ does not change as $x$ varies in each open simplex $\dot{\Delta}^{p}_{\sigma}$. So $E^{1}_{p,q}$ in the spectral sequence is isomorphic to

$$\bigoplus_{\sigma\in O_{p}}H_{p+q}((\Delta^{p}/\partial\Delta^{p})\wedge\eta^{-1}(b_{\sigma});\mathbb{Z}),$$

where $b_{\sigma}$ is the barycenter of the open simplex $\dot{\Delta}^{p}_{\sigma}$. Hence, given 2.5, the homology groups of the filtration quotients $\mathcal{G}_{p}/\mathcal{G}_{p-1}$ are finitely generated and concentrated in finitely many degrees which implies 1.1.

In the following sections, we find a model for $\eta^{-1}(x)$ to which we can apply the induction hypothesis (i.e. the homological finiteness of $\mathrm{B}\text{{Homeo}}(M,\text{rel }\partial)$ for $M$ with fewer prime factors) when $M$ is connected sum of irreducible factors that each has a non-trivial boundary. And we end this section with the proof of the contractibility of $\mathcal{S}(M)$.

In fact, it is a weakly Cohen-Macaulay complex of dimension infinity. Recall that a simplicial complex $X$ is called weakly Cohen-Macaulay of dimension $n$ if it is $(n-1)$-connected and the link of any $p$-simplex is $(n-p-2)$-connected. In this case, we denote this property by $\text{wCM}(X)\geq n$ (see [HW10, Definition 3.4]).

In this section, we shall use the hypothesis that the prime decomposition of $M$ consists of irreducible factors that each has non-empty boundary. The goal is for each $p$ and each $x\in\mathcal{F}_{p}-\mathcal{F}_{p-1}$ to find a semi-simplicial space $X_{\bullet}$ whose realization is homology equivalent to $\eta^{-1}(x)$ and sits in a fibration sequence so that by induction on the number of prime factors we could argue that the fiber and the base have finite CW complex model.

The advantage of working with $3$ manifolds that are connected sum of irreducible pieces such that each has a non-empty boundary is:

• •

  When we cut along essential separating spheres, the remaining pieces each has a non-spherical boundary component that is fixed and we shall use this for the inductive argument.

• •

  Since each irreducible factor has a non-trivial boundary that is fixed, homeomorphic irreducible factors cannot be permuted under the action of $\text{{Homeo}}(M,\text{rel }\partial)$.

To start, let $x$ be in $\mathcal{F}_{0}$ in the filtration 2.1 which is the image of a separatings sphere $S\subset M$. Let $\mathcal{S}(M,[S])$ be the full subcomplex of $\mathcal{S}(M)$ whose vertices are the orbits of $S$ under the action of $\text{{Homeo}}^{\delta}(M,\text{rel }\partial)$. Note that the preimage of $\eta^{-1}(x)$ is homotopy equivalent to

$$\mathcal{S}(M,[S])/\!\!/\text{{Homeo}}^{\delta}(M,\text{rel }\partial).$$

Suppose that $S$ cuts the manifold $M$ into two pieces $M_{1}$ and $M_{2}$ where $M_{1}$ contains $\partial_{*}M$, the boundary component of $M$ with the base point. . And let $\text{{Homeo}}(M_{1},S,\text{rel }\partial M)$ be the subgroup of $\text{{Homeo}}(M_{1})$ that fixes the boundary component $S$ set-wise and the rest of the boundary components pointwise. Towards our goal in this section, we prove that $\eta^{-1}(x)$ is homology equivalent to the realization of a semi-simplicial space (in fact a two-sided bar construction) $X_{\bullet}$ that fits in a fibration

(3.1)

$$\mathrm{B}\text{{Homeo}}(M_{2},\text{rel }\partial)\to||X_{\bullet}||\to\mathrm{B}\text{{Homeo}}(M_{1},S,\text{rel }\partial M).$$

By induction the fiber $\mathrm{B}\text{{Homeo}}(M_{2},\text{rel }\partial)$ has a finite CW complex model and in 3.9, we show that the base also has a finite CW complex model. First, we note that there is a natural order on vertices of each simplex in $\mathcal{S}(M,[S])$.

Now we define a semi-simplicial space whose underlying semi-simplicial set is $\mathcal{S}_{\bullet}(M,[S])$.

Since the action of $\text{{Homeo}}^{\delta}(M,\text{rel }\partial)$ on the set of $p$-simplices $\mathcal{S}_{p}(M,[S])$ is transitive, it is easy to use Thurston’s homology isomorphism 1.1 to obtain that for each $p$, the natural map

$$\mathcal{S}_{p}(M,[S])/\!\!/\text{{Homeo}}^{\delta}(M,\text{rel }\partial)\to\mathcal{S}^{\tau}_{p}(M,[S])/\!\!/\text{{Homeo}}(M,\text{rel }\partial),$$

induces a homology isomorphism. Therefore, by the spectral sequence that calculates the homology of the realization (see [ERW19, Section 1.4]), we have a homology isomorphism between fat realizations

$$||\mathcal{S}_{\bullet}(M,[S])/\!\!/\text{{Homeo}}^{\delta}(M,\text{rel }\partial)||\to||\mathcal{S}^{\tau}_{\bullet}(M,[S])/\!\!/\text{{Homeo}}(M,\text{rel }\partial)||$$

Hence, to prove homological finiteness for $\eta^{-1}(x)$, we find a model for

$$||\mathcal{S}^{\tau}_{\bullet}(M,[S])/\!\!/\text{{Homeo}}(M,\text{rel }\partial)||,$$

that sits in a fibration 3.1.

We can define the smooth version of $\mathcal{S}^{\tau}_{\bullet}(M,[S])$ and work with diffeomorphism groups. But in this dimension and for codimension $1$ embeddings, the corresponding objects in the $C^{0}$ and $C^{\infty}$-category are weakly homotopy equivalent. So we stick to $C^{0}$-category.