[Paper Review] Homological stability for moduli spaces of high dimensional manifolds

Søren Galatius, Oscar Randal‐Williams|arXiv (Cornell University)|Mar 30, 2012

Homotopy and Cohomology in Algebraic TopologyMathematics21 citations

TL;DR

This paper establishes homological stability for moduli spaces of high-dimensional manifolds diffeomorphic to connected sums of $S^n \times S^n$, proving that the inclusion of such manifolds induces isomorphisms in homology up to degree $k \leq (g-4)/2$ for $n > 2$. The result generalizes Harer's stability theorem for surface mapping class groups and enables a computation of the stable rational cohomology of these moduli spaces via a higher-dimensional analogue of Mumford's conjecture.

ABSTRACT

We prove a homological stability theorem for the moduli spaces of manifolds diffeomorphic to g(S^n x S^n), provided n > 2. This generalises 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 these moduli spaces in a range of degrees.

Motivation & Objective

• To establish a homological stability theorem for moduli spaces of $2n$-dimensional manifolds diffeomorphic to $\#^g S^n \times S^n$, generalizing Harer's result for surfaces.
• To extend the Madsen–Weiss theorem to higher-dimensional manifolds by identifying the stable homology of the moduli space $\mathscr{M}_\infty^n$ with that of an infinite loop space $\Omega^\infty MT\theta^n$.
• To compute the rational cohomology of the moduli spaces $\mathscr{M}_g^n$ in a stable range using generalized MMM classes.
• To provide a higher-dimensional analogue of Mumford's conjecture on the tautological ring of moduli spaces of curves.
• To prove that the stabilisation map $H_k(\mathscr{M}_g) \to H_k(\mathscr{M}_{g+1})$ is an isomorphism for $k \leq (g-4)/2$ when $n > 2$.

Proposed method

• The authors use an augmented simplicial space $X_\bullet$ built from embeddings of $W_{g,1} = \#^g(S^n \times S^n) - \text{int}(D^{2n})$ into $[0,\infty) \times \mathbb{R}^N$, with face maps induced by gluing in a standard cobordism.
• They analyze the spectral sequence associated to this simplicial space, with $E^1_{p,q} = H_q(X_p) \cong H_q(\mathscr{M}_{g-p-1})$, and show that the differential $d^1$ is isomorphic to the stabilisation map when $p$ is even and zero when $p$ is odd.
• The key step uses connectivity estimates on the realization $|X_\bullet| \to X_{-1}$, showing that the relative homology vanishes in degrees $\leq \lfloor(g-3)/2\rfloor$, which controls the spectral sequence.
• The proof proceeds by induction on $g$, using the vanishing of $E^\infty_{p,q}$ for $p+q \leq \lfloor(g-5)/2\rfloor$ to deduce that the stabilisation map is an isomorphism in the claimed range.
• The construction relies on the Pontrjagin–Thom construction to define a map $\alpha: \mathscr{M}_\infty^n \to \Omega^\infty MT\theta^n$, which induces an isomorphism on homology in the basepoint component.
• Generalized MMM classes $\kappa_c$ are defined via integration of differential forms pulled back from Grassmannians, and are shown to generate the stable rational cohomology.

Experimental results

Research questions

• RQ1 Does the stabilisation map $H_k(\mathscr{M}_g) \to H_k(\mathscr{M}_{g+1})$ become an isomorphism for $k \leq (g-4)/2$ in the moduli space of $2n$-manifolds $\#^g S^n \times S^n$ when $n > 2$?
• RQ2 Can the stable rational cohomology of $\mathscr{M}_g^n$ be computed in a range of degrees, analogous to Mumford’s conjecture for surfaces?
• RQ3 Is there a higher-dimensional analogue of the Madsen–Weiss theorem that identifies the stable homology of $\mathscr{M}_\infty^n$ with that of an infinite loop space $\Omega^\infty MT\theta^n$?
• RQ4 Do generalized MMM classes $\kappa_c$ defined via integration of characteristic classes on Grassmannians generate the stable rational cohomology of $\mathscr{M}_g^n$?
• RQ5 How does the spectral sequence of the augmented simplicial space $X_\bullet$ control the homological stability of $\mathscr{M}_g^n$?

Key findings

• The stabilisation map $H_k(\mathscr{M}_g) \to H_k(\mathscr{M}_{g+1})$ is an isomorphism for $k \leq (g-4)/2$ when $n > 2$, establishing homological stability for the moduli spaces of high-dimensional manifolds.
• The stable rational cohomology of $\mathscr{M}_g^n$ is isomorphic to the polynomial algebra generated by generalized MMM classes $\kappa_c$ for $c \in H^{*}(BSO(2n);\mathbb{Q})$, as stated in Corollary 1.3.
• The map $\alpha: \mathscr{M}_\infty^n \to \Omega^\infty MT\theta^n$ induces an isomorphism on homology in the basepoint component, generalizing the Madsen–Weiss theorem to higher dimensions.
• The spectral sequence of the augmented simplicial space $X_\bullet$ has $E^1_{p,q} \cong H_q(\mathscr{M}_{g-p-1})$, and the differential $d^1$ is isomorphic to the stabilisation map when $p$ is even and zero when $p$ is odd.
• For $n = 3$ or $n = 7$, the stabilisation map is an isomorphism for $g \geq 2k+2$ and surjective for $g \geq 2k$, due to improved connectivity of the relevant quadratic module space.
• The generalized MMM classes $\kappa_c$ are defined via integration of pullbacks of differential forms from $\mathrm{Gr}_{2n}^+(\mathbb{R}^{N+1})$, and define a map $H^{k+2n}(BSO(2n);\mathbb{R}) \to H^k(\mathscr{M}_g^n;\mathbb{R})$.