Homological stability for moduli spaces of high dimensional manifolds. I

Abstract

We prove a homological stability theorem for moduli spaces of simply connected manifolds of dimension $2n > 4$, with respect to forming connected sum with $S^n \times S^n$. This is analogous to Harer’s stability theorem for the homology of mapping class groups. Combined with previous work of the authors, it gives a calculation of the homology of the moduli spaces of manifolds diffeomorphic to connected sums of $S^n \times S^n$ in a range of degrees.

1. Introduction and statement of results

A famous result of Harer Har85 established homological stability for mapping class groups of oriented surfaces. For example, if $\Gamma _{g,1}$ denotes the group of isotopy classes of diffeomorphisms of an oriented connected surface of genus $g$ with one boundary component, then the natural homomorphism $\Gamma _{g,1} \to \Gamma _{g+1,1}$, given by gluing on a genus one surface with two boundary components, induces an isomorphism in group homology $H_k(\Gamma _{g,1}) \to H_k(\Gamma _{g+1,1})$ as long as $k \leq (2g-2)/3$. (Harer proved this for $k \leq (g+1)/3$, but the range was later improved by Ivanov Iva93 and Boldsen Bol12; see also RW16.) This result can be interpreted in terms of moduli spaces of Riemann surfaces, and it has lead to a wealth of research in topology and algebraic geometry. In this paper we will prove an analogous homological stability result for moduli spaces of manifolds of higher (even) dimension.

Definition 1.1.

For a compact smooth manifold $W$, let $\mathrm{Diff}_\partial (W)$ denote the topological group of diffeomorphisms of $W$ restricting to the identity near its boundary. The moduli space of manifolds of type $W$ is defined as the classifying space $\mathscr{M}(W) = B\mathrm{Diff}_\partial (W)$.

If we are given another compact smooth manifold $W'$ and a codimension zero embedding $W \hookrightarrow W'\setminus \partial W'$, then we obtain a continuous homomorphism $\mathrm{Diff}_\partial (W) \to \mathrm{Diff}_\partial (W')$ by extending diffeomorphisms of $W$ by the identity diffeomorphism on the cobordism $K = W' \setminus \mathrm{int}(W)$. The induced map of classifying spaces shall be denoted

$$\begin{equation} - \cup K : \mathscr{M}(W) \longrightarrow \mathscr{M}(W \cup _{\partial W} K). {\tag{1.1}} \end{equation}$$

We shall give point-set models for $\mathscr{M}(W)$ and the map (1.1) in Section 6.

When $W$ is an orientable surface of genus $g$ with one boundary component, and $W' = W \cup _{\partial W} K$ is also orientable of genus $g+1$ with one boundary component, it can be shown that the map 1.1 is equivalent to the map studied by Harer, and hence it induces an isomorphism on homology in a range of degrees which increases with the genus of the surface. Our main result is analogous to this, but for simply connected manifolds of higher even dimension (although we exclude the case $2n=4$ for the usual reason: we shall need to use the Whitney trick). We must first describe the analogue of genus which we will use.

In each dimension $2n$ we define manifolds

$$\begin{equation*} W_{g,1} = W_{g,1}^{2n} = D^{2n} \sharp g(S^n \times S^n), \end{equation*}$$

the connected sum of $g$ copies of $S^n \times S^n$ with an open disc removed, and if $W$ is a compact path-connected $2n$-manifold, we define the number

$$\begin{equation*} g(W) = \max \{g \in \mathbb{N}\,\,\vert \,\, \text{there exists an embedding $W_{g,1} \hookrightarrow W$}\}, \end{equation*}$$

which we call the genus of $W$. Let $S$ be a manifold obtained by forming the connected sum of $[0,1] \times \partial W$ with $S^n \times S^n$. The corresponding gluing map shall be denoted

$$\begin{equation*} s = - \cup S: \mathscr{M}(W) \longrightarrow \mathscr{M}(W \cup _{\partial W} S). \end{equation*}$$

(If $\partial W$ is not path-connected, then the diffeomorphism type of $S$ relative to $\{0\} \times \partial W$, and hence the homotopy class of $s$, will depend on which path component the connected sum is formed in. The following theorem holds for any such choice.)

Theorem 1.2.

For a simply connected manifold $W$ of dimension $2n \geq 6$, the stabilisation map

$$\begin{equation*} s_*: H_k(\mathscr{M}(W)) \longrightarrow H_k(\mathscr{M}(W \cup _{\partial W} S)) \end{equation*}$$

is an isomorphism if $2k \leq g(W)-3$ and an epimorphism if $2k \leq g(W)-1$.

Our methods are similar to those used to prove many homological stability results for homology of discrete groups, namely to use a suitable action of the group on a simplicial complex. For example, Harer used the action of the mapping class group on the arc complex to prove his homological stability result. In our case the relevant groups are not discrete, so we use a semisimplicial space instead.

1.1. Tangential structures and abelian coefficients

In Section 7 we shall generalise Theorem 1.2 in two directions. First, we shall establish a version of Theorem 1.2 where $\mathscr{M}(W)$ is replaced by a space of manifolds equipped with certain tangential structures; second, we shall allow certain nontrivial systems of local coefficients.

Let $\gamma _{2n} \to BO(2n)$ denote the universal vector bundle. A tangential structure for $2n$-dimensional manifolds is a fibration $\theta : B \to BO(2n)$ with $B$ path-connected, classifying a vector bundle $\theta ^*\gamma _{2n}$ over $B$. Examples include $B = EO(2n)/G \to BO(2n)$ for $G = SO(2n)$, corresponding to an orientation, $G = U(n)$, corresponding to an almost complex structure, or $G = \{1\}$, corresponding to a framing. A $\theta$-structure on a $2n$-dimensional manifold is a map of vector bundles $\hat{\ell }:TW \to \theta ^*\gamma _{2n}$, and we then write $\ell : W \to B$ for the underlying map of spaces. If $\hat{\ell }_{\partial W} : TW\vert _{\partial W} \to \theta ^*\gamma _{2n}$ is a given $\theta$-structure, then we write $\mathrm{Bun}_\partial ^\theta (W;\hat{\ell }_{\partial W})$ for the space of bundle maps extending $\hat{\ell }_{\partial W}$. The group $\mathrm{Diff}_\partial (W)$ acts on this space, and we write

$$\mathscr{M}^\theta (W;\hat{\ell }_{\partial W}) = \mathrm{Bun}_\partial ^\theta (W;\hat{\ell }_{\partial W}) /\!\!/\mathrm{Diff}_\partial (W)$$

for the Borel construction. This space need not be path-connected, but if $\hat{\ell }_W \in \mathrm{Bun}_\partial ^\theta (W;\hat{\ell }_{\partial W})$, then we write $\mathscr{M}^\theta (W,\hat{\ell }_{W})$ for the path component of $\hat{\ell }_W$.

Our generalisation of Theorem 1.2 to manifolds with tangential structures will replace the spaces $\mathscr{M}(W)$ by $\mathscr{M}^\theta (W,\hat{\ell }_{W})$. Before stating the theorem, we must explain the corresponding notion of genus and the analogue of the cobordism $S$.

Definition 1.3.

Let us say that a $\theta$-structure $\hat{\ell } : TW_{1,1} \to \theta ^*\gamma _{2n}$ is admissible if there is a pair of orientation-preserving embeddings $e, f : S^n \times D^n \hookrightarrow W_{1,1} \subset S^n \times S^n$ whose cores $e(S^n \times \{0\})$ and $f(S^n \times \{0\})$ intersect transversely in a single point, such that each of the $\theta$-structures $e^*\hat{\ell }$ and $f^*\hat{\ell }$ on $S^n \times D^n$ extend to $\mathbb{R}^{2n}$ for some orientation-preserving embeddings $S^n \times D^n \hookrightarrow \mathbb{R}^{2n}$. We then define the $\theta$-genus of a compact path-connected $\theta$-manifold $(W, \hat{\ell }_W)$ to be

$$\begin{equation*} g^\theta (W, \hat{\ell }_W) = \max \left\{g \in \mathbb{N}\,\,\bigg |\,\, \text{there are $g$ disjoint copies of $W_{1,1}$ in $(W, \hat{\ell }_W)$, each with admissible $\theta $-structure}\right\}. \end{equation*}$$

If $W$ contains a copy of $W_{g,1}$ such that $\ell _W\vert _{W_{g,1}} : W_{g,1} \to B$ is nullhomotopic, then in Proposition 7.12 we show that $g^\theta (W, \hat{\ell }_W) \geq g-1$, and if $n \neq 3,7$, this can be strengthened to $g^\theta (W, \hat{\ell }_W) \geq g$. When $B$ is simply connected, the number $g^\theta (W,\hat{\ell }_W)$ may be estimated in terms of characteristic numbers, with a constant error term depending only on $n$ and $H_n(B;\mathbb{Z})$; cf. Remark 7.16.

In order to define the stabilisation map, we say that a $\theta$-structure $\hat{\ell }_S$ on $S$ is admissible if it is admissible in the sense above when restricted to $W_{1,1} \subset S$. Suppose furthermore that it restricts to $\hat{\ell }_{\partial W}$ on $\{0\} \times \partial W$, and write $\hat{\ell }_{\partial W}^\prime$ for its restriction to $\{1\} \times \partial W$. Then there is an induced stabilisation map

$$s = - \cup (S, \hat{\ell }_S) : \mathscr{M}^\theta (W,\hat{\ell }_{W}) \longrightarrow \mathscr{M}^\theta (W \cup _{\partial W} S,\hat{\ell }_{W} \cup \hat{\ell }_S)$$

given by gluing on $S$ to $W$ and extending $\theta$-structures using $\hat{\ell }_S$.

We require two additional terms to describe our result: we say that $\theta$ is spherical if $S^{2n}$ admits a $\theta$-structure, and we say that a local coefficient system is abelian if it has trivial monodromy along all nullhomologous loops.

Theorem 1.4.

For a simply connected manifold $W$ of dimension $2n \geq 6$, a $\theta$-structure $\hat{\ell }_W$ on $W$, an admissible $\theta$-structure $\hat{\ell }_S$ on $S$, and an abelian local coefficient system $\mathcal{L}$ on $\mathscr{M}^\theta (W \cup _{\partial W} S,\hat{\ell }_{W \cup S})$, the stabilisation map

$$\begin{equation*} s_*: H_k(\mathscr{M}^\theta (W,\hat{\ell }_{W});s^*\mathcal{L}) \longrightarrow H_k(\mathscr{M}^\theta (W \cup _{\partial W} S,\hat{\ell }_{ W \cup S});\mathcal{L}) \end{equation*}$$

is

(i)

an epimorphism for $3k \leq g^\theta (W, \hat{\ell }_W)-1$ and an isomorphism for $3k \leq g^\theta (W, \hat{\ell }_W)-4$,

(ii)

an epimorphism for $2k \leq g^\theta (W, \hat{\ell }_W)-1$ and an isomorphism for $2k \leq g^\theta (W, \hat{\ell }_W)-3$, if $\theta$ is spherical and $\mathcal{L}$ is constant.

For example, consider the tangential structure $\theta : BU(3) \to BO(6)$. If $(W, \hat{\ell }_W)$ is an almost complex 6-manifold (with non-empty boundary), and $e : W_{g,1} \to W$ is an embedding, then $\ell _W \circ e : W_{g,1} \to BU(3)$ is nullhomotopic because $W_{g,1} \simeq \vee ^{2g} S^3$ and $\pi _3(BU(3))=0$. Thus $g^\theta (W, \hat{\ell }_W) \geq g-1$. Furthermore, $S^6$ admits an almost complex structure so $\theta$ is spherical. So for any admissible $\theta$-structure $\hat{\ell }_S$ on $S$, the stabilisation map

$$s : \mathscr{M}^\theta (W,\hat{\ell }_{W}) \longrightarrow \mathscr{M}^\theta (W \cup _{\partial W} S,\hat{\ell }_{ W \cup S})$$

induces an isomorphism on integral homology in degrees up to $\tfrac{g-4}{2}$.

In the sequel GRW16b to this paper we prove an analogue of Theorem 1.4 where the manifold $S$ is replaced by a more general cobordism $K$, satisfying that $(K,\partial W)$ is $(n-1)$-connected. The theorem proved there includes the case where $W \cup _{\partial W} K$ is a closed manifold.

1.2. Stable homology

If we have a sequence $\hat{\ell }_{S_1}, \hat{\ell }_{S_2}, \ldots$ of admissible $\theta$-structures on $S$ such that $\hat{\ell }_{S_1}\vert _{\{0\} \times \partial W} = \hat{\ell }_W\vert _{\partial W}$ and $\hat{\ell }_{S_i}\vert _{\{1\} \times \partial W} = \hat{\ell }_{S_{i+1}}\vert _{\{0\} \times \partial W}$ for all $i$, then the manifold $W \cup gS$ given by the composition of $W$ and $g$ copies of the cobordism $S$ has a $\theta$-structure $\hat{\ell }_{W \cup gS} = \hat{\ell }_W \cup \hat{\ell }_{S_1} \cup \cdots \cup \hat{\ell }_{S_g}$, and there are maps

$$- \cup (S, \hat{\ell }_{S_{g+1}}) : \mathscr{M}^\theta (W \cup gS, \hat{\ell }_{W \cup gS}) \longrightarrow \mathscr{M}^\theta (W \cup (g+1)S, \hat{\ell }_{W \cup (g+1)S}).$$

In this situation the homology of the limiting space

$$\operatorname *{hocolim}_{g \to \infty } \mathscr{M}^\theta (W \cup gS, \hat{\ell }_{W \cup gS})$$

can be described in homotopy-theoretic terms for any $W$ and any $\theta$, from which explicit calculations are quite feasible. In many cases this description is given in GRW14b, and we shall describe the general case in GRW16b, Sections 1.2 and 7. Here we focus on the interesting special case $W = D^{2n}$ and $\theta = \mathrm{Id}_{BO(2n)}$, in which case $\mathscr{M}^\theta (W \cup gS , \hat{\ell }_{W \cup gS}) = \mathscr{M}(W_{g,1}) = B\mathrm{Diff}_\partial (W_{g,1})$.

The boundary of $W_{g,1}$ is a sphere, so $S = ([0,1] \times S^{2n-1}) \sharp (S^n \times S^n)$, and hence there is a diffeomorphism $W_{g,1} \cup _{\partial W_{g,1}} S \approx W_{g+1,1}$ relative to their already identified boundaries. Theorem 1.2 (or Theorem 1.4 for abelian coefficients) therefore implies the following.

Corollary 1.5.

For $2n \geq 6$ and an abelian coefficient system $\mathcal{L}$ on $\mathscr{M}(W_{g+1,1})$, the stabilisation map

$$\begin{equation*} s_* : H_k(\mathscr{M}(W_{g,1});s^*\mathcal{L}) \longrightarrow H_k(\mathscr{M}(W_{g+1,1});\mathcal{L}) \end{equation*}$$

is an epimorphism for $3k \leq g-1$ and an isomorphism for $3k \leq g-4$. If $\mathcal{L}$ is constant, then it is an epimorphism for $2k \leq g-1$ and an isomorphism for $2k \leq g-3$.

It is an immediate consequence of this corollary and RW13, Section 3 that the mapping class group $\pi _1(\mathscr{M}(W_{g,1}))$ has perfect commutator subgroup as long as $g \geq 7$.

Remark 1.6.

Independently, Berglund and Madsen BM13 have obtained a result similar to this corollary, for rational cohomology in the range $2k \leq \min (2n-6, g-6)$.

Remark 1.7.

In the earlier preprint GRW12 we considered only the manifolds $W_{g,1}$, rather than the more general manifolds of Theorem 1.2. Although the present paper entirely subsumes GRW12, the reader mainly interested in Corollary 1.5 may want to consult the preprint for a streamlined text adapted to that special case (at least if they are only concerned with constant coefficients).

By the universal coefficient theorem, stability for homology implies stability for cohomology; in the surface case, Mumford Mum83 conjectured an explicit formula for the stable rational cohomology, which in our notation asserts that a certain ring homomorphism

$$\begin{equation*} \mathbb{Q}[\kappa _1, \kappa _2, \dots ] \longrightarrow H^*(\mathscr{M}(W^2_{g,1});\mathbb{Q}) \end{equation*}$$

is an isomorphism for $g \gg *$. Mumford’s conjecture was proved in a strengthened form by Madsen and Weiss MW07.

Corollary 1.5 and our previous paper GRW14b allow us to prove results analogous to Mumford’s conjecture and the Madsen–Weiss theorem for the moduli spaces $\mathscr{M}(W_{g,1})$ with $2n \geq 6$. The analogue of the Madsen–Weiss theorem for these spaces concerns the homology of the limiting space $\mathscr{M}(W_\infty ) = \operatorname *{hocolim}\limits _{g \to \infty } \mathscr{M}(W_{g,1})$. There is a certain infinite loop space $\Omega ^\infty MT\theta ^n$ associated to the tangential structure $\theta ^n : BO(2n)\langle n \rangle \to BO(2n)$ given by the $n$-connected cover, and a map

$$\alpha : \mathscr{M}(W_\infty ) \longrightarrow \Omega ^\infty MT\theta ^n$$

given by a parametrised form of the Pontryagin–Thom construction. In GRW14b, Theorem 1.1 we proved that $\alpha$ induces an isomorphism between the homology of $\mathscr{M}(W_\infty )$ and the homology of the basepoint component of $\Omega ^\infty MT\theta ^n$. In GRW16 we used this to compute $H_1(\mathscr{M}(W_{g,1});\mathbb{Z})$ for $g \geq 5$.

It is easy to calculate the rational cohomology ring of a component of $\Omega ^\infty MT\theta ^n$, and hence of $\mathscr{M}(W_{g,1})$ in the range of degrees given by Corollary 1.5. The result is Corollary 1.8 below, which is a higher-dimensional analogue of Mumford’s conjecture. Recall that for each $c \in H^{k+2n}(BSO(2n))$ there is an associated cohomology class $\kappa _c \in H^{k}(\Omega ^\infty MT\theta ^n)$. Pulling it back via $\alpha$ and all the stabilisation maps $\mathscr{M}(W_{g,1}) \to \mathscr{M}(W_{\infty })$ defines classes $\kappa _c \in H^k(\mathscr{M}(W_{g,1}))$ for all $g$, sometimes called “generalised Mumford–Morita–Miller classes”. These can also be described in terms of fibrewise integration; see e.g. GRW14a, Section 1.1. The following result is our higher-dimensional analogue of Mumford’s conjecture.

Corollary 1.8.

Let $2n \geq 6$, and let $\mathcal{B}\subset H^*(BSO(2n);\mathbb{Q})$ be the set of monomials in the classes $e, p_{n-1}, \dots , p_{\lceil \frac{n+1}{4} \rceil }$, of degree greater than $2n$. Then the induced map

$$\begin{equation*} \mathbb{Q}[\kappa _c \,\,|\,\, c \in \mathcal{B}] \longrightarrow H^*(\mathscr{M}(W_{g,1});\mathbb{Q}) \end{equation*}$$

is an isomorphism in degrees satisfying $2* \leq g-3$.

For example, if $2n = 6$, the set $\mathcal{B}$ consists of monomials in $e$, $p_1$, and $p_2$, and therefore $H^*(\mathscr{M}(W_{g,1}^{6});\mathbb{Q})$ agrees for $\ast \leq \tfrac{g-3}{2}$ with a polynomial ring in variables of degrees 2, 2, 4, 6, 6, 6, 8, 8, 10, 10, 10, 10, 12, 12, ….

2. Techniques

In this section we collect the technical results needed to establish high connectivity of certain simplicial spaces which will be relevant for the proof of Theorem 1.2. The main results are Theorem 2.4 and Corollary 2.9.

2.1. Cohen–Macaulay complexes

Recall from HW10, Definition 3.4 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 write $\mathrm{wCM}(X) \geq n$. We shall also say that $X$ is locally weakly Cohen–Macaulay of dimension $n$ if the link of any $p$-simplex is $(n-p-2)$-connected (but no global connectivity is required on $X$ itself). In this case we shall write $\mathrm{lCM}(X) \geq n$.

Lemma 2.1.

If $\mathrm{lCM}(X) \geq n$ and $\sigma < X$ is a $p$-simplex, then $\mathrm{wCM}(\mathrm{Lk}(\sigma )) \geq n-p-1$.

Proof.

By assumption, $\mathrm{Lk}(\sigma )$ is $((n-p-1)-1)$-connected. If $\tau < \mathrm{Lk}(\sigma )$ is a $q$-simplex, then

$$\begin{equation*} \mathrm{Lk}_{\mathrm{Lk}(\sigma )}(\tau ) = \mathrm{Lk}_X(\sigma \ast \tau ) \end{equation*}$$

is $((n-p-1) - q - 2)$-connected, since $\sigma \ast \tau$ is a $(p+q+1)$-simplex, and hence its link in $X$ is $(n-(p+q+1)-2)$-connected.

■

Definition 2.2.

Let us say that a simplicial map