Stability Properties of Moduli Spaces

1. Moduli Spaces and Stability

Moduli spaces are spaces that parameterize topological or geometric data. They often appear in families; for example, the configuration spaces of $n$ points in a fixed manifold, the Grassmannians of linear subspaces of dimension $d$ in $\mathbb{R}^{\infty }$, and the moduli spaces $\mathcal{M}_g$ of Riemann surfaces of genus $g$. These families are usually indexed by some geometrically defined quantity, such as the number $n$ of points in a configuration, the dimension $d$ of the linear subspaces, or the genus $g$ of a Riemann surface. Understanding the topology of these spaces has been a subject of intense interest for the last 60 years.

For a family of moduli spaces $\{X_n\}_n$ we ask:

Question 1.1.

How does the topology of the moduli spaces $X_n$ change as the parameter $n$ changes?

For many natural examples of moduli spaces $X_n$, some aspects of the topology get more complicated as the parameter $n$ gets larger. For instance, the dimension of $X_n$ frequently increases with $n$ as well as the number of generators and relations needed to give a presentation of their fundamental groups. But, maybe surprisingly, there are sometimes features of the moduli spaces that ‘stabilize’ as $n$ increases. In this survey we will describe some forms of stability and some examples of where they arise.

1.1. Homology and cohomology

Algebraic topology is a branch of mathematics that uses tools from abstract algebra to classify and study topological spaces. By constructing algebraic invariants of topological spaces, we can translate topological problems into (typically easier) algebraic ones. An algebraic invariant of a space is a quantity or algebraic object, such as a group, that is preserved under homeomorphism or homotopy equivalence. One example is the fundamental group $\pi _1(X,x_0)$ of homotopy classes of loops in a topological space $X$ based at the point $x_0$. Homology and cohomology groups are other examples and are the focus of this article. Their definitions are more subtle than those of homotopy groups like $\pi _1(X,x_0)$, but they are often more computationally tractable and are widely studied.

Given a topological space $X$ and $k \in \mathbb{Z}_{\geq 0}$, we can associate groups $H_k (X;R)$ and $H^k(X;R)$, the $k$th homology and cohomology groups (with coefficients in $R$), where $R$ is a commutative ring such as $\mathbb{Z}$ or $\mathbb{Q}$. These algebraic invariants define functors from the category of topological spaces to the category of $R$-modules: for any continuous map of topological spaces $f \colon X\rightarrow Y$ there are induced $R$-linear maps

$$f_*\colon H_k(X;R)\rightarrow H_k(Y;R)\text{ (covariant),}$$

$$f^*\colon H^k(Y;R)\rightarrow H^k(X;R)\text{ (contravariant)}.$$ The cohomology groups $H^*(X;R) = \bigoplus _k H^k(X; R)$ in fact have the structure of a graded $R$-algebra with respect to the cup product operation.

The group $H_0(X;\mathbb{Z})$ is the free abelian group on the path components of the topological space $X$ and $H^0(X; \mathbb{Z})$ is its dual. If $X$ is path-connected, $H_1(X;\mathbb{Z})$ is naturally isomorphic to the abelianization of $\pi _1(X, x_0)$ with respect to any basepoint $x_0$, and its elements are certain equivalence classes of (unbased) loops in $X$.

For a topological group $G$ there exists an associated classifying space $\text{B}G$ for principal $G$-bundles. It is constructed as the quotient of a (weakly) contractible space $\text{E}G$ by a proper free action of $G$. The space $\text{B}G$ is unique up to (weak) homotopy equivalence. If $G$ is a discrete group, then $\text{B}G$ is precisely an Eilenberg-MacLane space $K(G,1)$, i.e., a path-connected topological space with $\pi _1(\text{B}G) \cong G$ and trivial higher homotopy groups. For example, up to homotopy equivalence, $\text{B}\mathbb{Z}$ is the circle, $\text{B}\mathbb{Z}_2$ is the infinite-dimensional real projective space $\mathbb{R}\mathrm{P}^{\infty }$, and the Grassmanian of $d$-dimensional linear subspaces in $\mathbb{R}^{\infty }$ is $\text{B}\text{GL}_d(\mathbb{R})$.

Some motivation to study the cohomology of $\text{B}G$: its cohomology classes define characteristic classes of principal $G$-bundles, invariants that measure the ‘twistedness’ of the bundle. For instance the cohomology algebra $H^*( \text{B}\text{GL}_d(\mathbb{R});\mathbb{Z})$ can be described in terms of Pontryagin and Stiefel–Whitney classes.

With $\text{B}G$ we can define the group homology and group cohomology of a discrete group $G$ by

$$H_k(G;R) \coloneq H_k(\text{B}G;R),\ \ H^k(G; R) \coloneq H^k(\text{B}G; R).$$

We can refine Question 1.1 to the following:

Question 1.2.

Given family $\{X_n\}_{n}$ of moduli spaces or discrete groups, how do the homology and cohomology groups of the $n$th space in the sequence change as the parameter $n$ increases?

In this article we discuss Question 1.2 with a particular focus on the families of configuration spaces and braid groups. For further reading⁠¹ we recommend R. Cohen’s survey Coh09 on stability of moduli spaces.

¹

A version of this note with an extended reference list is available at https://arxiv.org/abs/2201.04096.

✖

1.2. Homological stability

Definition 1.3.

A sequence of spaces or groups $\{X_n\}_{n \geq 0}$ with maps

$$X_{0}\xrightarrow {s_{0}} \ldots \xrightarrow {s_{n-2}} X_{n-1}\xrightarrow {s_{n-1}} X_n\xrightarrow {s_n} X_{n+1}\xrightarrow {s_{n+1}}\ldots$$

satisfies homological stability if, for each $k$, the induced map in degree-$k$ homology

$$(s_n)_*\colon H_k(X_n;\mathbb{Z})\rightarrow H_k(X_{n+1};\mathbb{Z})$$

is an isomorphism for all $n \geq N_k$ for some stability threshold $N_k \in \mathbb{Z}$ depending on $k$. The maps $s_n$ are sometimes called stabilization maps and the set $\{ (n,k) \in \mathbb{Z}^2 \; | \; n \geq N_k\}$ is the stable range.

If the maps $s_n\colon X_n \to X_{n+1}$ are inclusions we define $X_{\infty } \coloneq \bigcup _{n\geq 1} X_n$ to be the stable group or space. Under mild assumptions, if $\{X_n\}_n$ satisfies homological stability, then

$$H_k(X_{\infty };\mathbb{Z})\cong H_k(X_n;\mathbb{Z}) \text{ for }n \geq N_k.$$

We call the groups $H_k(X_{\infty };\mathbb{Z})$ the stable homology.

2. An Example: Configuration Spaces and the Braid Groups

2.1. A primer on configuration spaces

Definition 2.1.

Let $M$ be a topological space, such as a graph or a manifold. The (ordered) configuration space $F_n(M)$ of $n$ particles on $M$ is the space

$$F_n(M) = \{ (x_1, \ldots , x_n) \in M^n \; | \; x_1, \ldots , x_n \text{ distinct} \},$$

topologized as a subspace of $M^n$. Notably, $F_0(M)$ is a point and $F_1(M)=M$.

Configuration spaces have a long history of study in connection to topics as broad-ranging as homotopy groups of spheres and robotic motion planning.

One way to conceptualize the configuration space $F_n(M)$ is as the complement of the union of subspaces of $M^n$ defined by equations of the form $x_i = x_j$.

Figure 1.

The space $F_2([0,1])$ is obtained by deleting the diagonal from the square $[0,1]^2$.

$F_2([0,1]) =$

In other words, we can construct $F_n(M)$ by deleting the “fat diagonal” of $M^n$, consisting of all $n$-tuples in $M^n$ where two or more components coincide. In the simplest case, when $n=2$ and $M$ is the interval $[0,1]$, we see that $F_2([0,1])$ consists of two contractible components, as in Figure 1.

Another way we can conceptualize $F_n(M)$ is as the space of embeddings of the discrete set $\{1, 2, \ldots , n\}$ into $M$, appropriately topologized. We may visualize a point in $F_n(M)$ by labelling $n$ points in $M$, as in Figure 2.

Figure 2.

A point in the ordered configuration space of an open surface $\Sigma$.

$\in F_4(\Sigma )$

From this perspective, we may reinterpret the path components of $F_2([0,1])$: one component consists of all configurations where particle 1 is to the left of particle 2, and one component has particle 1 on the right. See Figure 3.

Figure 3.

The path components of $F_2([0,1])$.

$F_2([0,1]) =$

Any path through $[0,1]^2$ that interchanges the relative positions of the two particles must involve a ‘collision’ of particles, and hence exit the configuration space $F_2([0,1]) \subseteq [0,1]^2$. We encourage the reader to verify that, in general, the configuration space $F_n([0,1])$ is the union of $n!$ contractible path components, indexed by elements of the symmetric group $S_n$. See Figure 4.

Figure 4.

A point in $F_4([0,1])$ in the path component indexed by the permutation $2143$ in $S_4$.

    $\in F_4([0,1])$

In contrast, if $M$ is a connected manifold of dimension $2$ or more, then $F_n(M)$ is path-connected: given any two configurations, we can construct a path through $M^n$ from one configuration to the other without any ‘collisions’ of particles. In this case $H_0(F_n(M);\mathbb{Z}) \cong \mathbb{Z}$ for all $n \geq 0$, and this is our first glimpse of stability in these spaces as $n \to \infty$.

For any space $M$, the symmetric group $S_n$ acts freely on $F_n(M)$ by permuting the coordinates of an $n$-tuple $(x_1, \ldots , x_n)$, equivalently, by permuting the labels on a configuration as in Figure 2. The orbit space $C_n(M) = F_n(M) /S_n$ is the (unordered) configuration space of $n$ particles on $M$. This is the space of all $n$-element subsets of $M$, topologized as the quotient of $F_n(M)$. The reader may verify that the quotient map (illustrated in Figure 5) is a regular $S_n$-covering space map. In particular, by covering space theory, the quotient map $F_n(M) \to C_n(M)$ induces an injective map on fundamental groups.

Figure 5.

The quotient map $F_n(M) \to C_n(M)$.

In the case that $M$ is the complex plane $\mathbb{C}$, we can identify $C_n(\mathbb{C})$ with the space of monic degree-$n$ polynomials over $\mathbb{C}$ with distinct roots, by mapping a configuration $\{z_1, \ldots , z_n\}$ to the polynomial $p(x)=(x-z_1)\cdots (x-z_n)$. For this reason the topology of $C_n(\mathbb{C})$ has deep connections to classical problems about finding roots of polynomials.

We will address Question 1.2 for the families $\{ C_n(M)\}_n$ and $\{ F_n(M)\}_n$, but we first specialize to the case when $M=\mathbb{C}$. Although the spaces $C_n(\mathbb{C})$ and $F_n(\mathbb{C})$ are path-connected, in contrast to the configuration spaces of $M=[0,1]$, they have rich topological structures: they are classifying spaces for the braid groups and the pure braid groups, respectively, which we now introduce.

2.2. A primer on the braid groups

Since $F_n(\mathbb{C})$ is path-connected, as an abstract group its fundamental group is independent of choice of basepoint. For path-connected spaces, we sometimes drop the basepoint from the notation for $\pi _1$.

Definition 2.2.

The fundamental group $\pi _1(C_n(\mathbb{C}))$ is called the braid group $\mathbf{B}_n$ and $\pi _1(F_n(\mathbb{C}))$ is the pure braid group $\mathbf{P}_n$.

We can understand $\pi _1(F_n(\mathbb{C}))$ as follows. Choose a basepoint configuration $(z_1, \ldots , z_n)$ in $F_n(\mathbb{C})$, and then we may visualize a loop as a ‘movie’ where the $n$ particles continuously move around $\mathbb{C}$, eventually returning pointwise to their starting positions. If we represent time by a third spacial dimension, as shown in Figure 6, we can view the particles as tracing out a braid. Note that, up to homeomorphism, we may view $F_n(\mathbb{C})$ as the configuration space of the open 2-disk.

Figure 6.

A visualization of a loop $\gamma (t)$ in $F_5(\mathbb{C})$ representing an element of $\pi _1(F_5(\mathbb{C})) \cong \mathbf{P}_5$.

Loops in $C_n(\mathbb{C})$ are similar, with the crucial distinction that the $n$ particles are unlabelled and indistinguishable, and so need only return set-wise to their basepoint configuration.

Figure 7.

A braid on 3 strands.

It is traditional to represent elements of the group $\mathbf{B}_n$ and its subgroup $\mathbf{P}_n$ by equivalence classes of braid diagrams, as illustrated in Figure 7. These braid diagrams depict $n$ strings (called strands) in Euclidean 3-space, anchored at their tops at $n$ distinguished points in a horizontal plane, and anchored at their bottoms at the same $n$ points in a parallel plane. The strands may move in space but may not double back or pass through each other. The group operation is concatenation, as in Figure 8.

Figure 8.

The group structure on $\mathbf{B}_n$.

The braid groups were defined rigorously by Artin in 1925, but the roots of this notion appeared in the earlier work of Hurwitz, Firckle, and Klein in the 1890s and of Vandermonde in 1771. This topological interpretation of braid groups as the fundamental groups of configuration spaces was formalized in 1962 by Fox and Neuwirth.

Artin established presentations for the braid group and the pure braid group. His presentation for $\mathbf{B}_n$,

$$\begin{equation*} \mathbf{B}_n \cong \left\langle \sigma _1,\sigma _2\dots , \sigma _{n-1} \biggm | \begin{array}{c} \sigma _i\sigma _j = \sigma _j\sigma _i \ \text{if} \ |i-j| \geq 2 \\\sigma _i\sigma _{i+1}\sigma _i = \sigma _{i+1}\sigma _i\sigma _{i+1} \end{array} \right\rangle , \end{equation*}$$

uses $(n-1)$ generators $\sigma _i$ corresponding to half-twists of adjacent strands, as in Figure 9.

Figure 9.

Artin’s generator $\sigma _i$ for $\mathbf{B}_n$.

Artin also gave a finite presentation for $\mathbf{P}_n$. We will not state it in full, but comment that there are ${n \choose 2}$ generators $T_{ij}$, ($i \neq j$, $i,j \in \{1, 2, \ldots , n\})$ corresponding to full twists of each pair of strands, as in Figure 10.

Figure 10.

Artin’s generator $T_{ij}=T_{ji}$ for $\mathbf{P}_n$.

Corresponding to the regular covering space map $F_n(\mathbb{C}) \to C_n(\mathbb{C})$ of Figure 5, there is a short exact sequence of groups

$$1 \to \mathbf{P}_n \to \mathbf{B}_n \to S_n \to 1.$$

The quotient map $\mathbf{B}_n \to S_n$, shown in Figure 11, takes a braid, forgets the $n$ strands and simply records the permutation induced on their endpoints. The generator $\sigma _i$ maps to the simple transposition $(i \; i+1)$. The kernel is those braids that induce the trivial permutation, i.e., the pure braid group.

Figure 11.

The quotient map $\mathbf{B}_n \to S_n$.

2.3. Homological stability for the braid groups

Arnold calculated some homology groups of $\mathbf{B}_n$ in low degree (Table 1).

Table 1.

The homology groups $H_k(\mathbf{B}_n;\mathbb{Z})$. Empty spaces are zero groups. Stable groups are shaded.

		$k$	0	1	2	3	4	5
	$n$
	0		$\mathbb{Z}$
	1		$\mathbb{Z}$
	2		$\mathbb{Z}$	${\mathbb{Z}}$
	3		$\mathbb{Z}$	$\mathbb{Z}$
	4		$\mathbb{Z}$	$\mathbb{Z}$	$\mathbb{Z}_2$
	5		$\mathbb{Z}$	$\mathbb{Z}$	$\mathbb{Z}_2$
	6		$\mathbb{Z}$	$\mathbb{Z}$	$\mathbb{Z}_2$	${ \mathbb{Z}_2}$	$\mathbb{Z}_3$
	7		$\mathbb{Z}$	$\mathbb{Z}$	$\mathbb{Z}_2$	$\mathbb{Z}_2$	$\mathbb{Z}_3$
	8		$\mathbb{Z}$	$\mathbb{Z}$	$\mathbb{Z}_2$	$\mathbb{Z}_2$	${\mathbb{Z}_6}$	$\mathbb{Z}_3$
	9		$\mathbb{Z}$	$\mathbb{Z}$	$\mathbb{Z}_2$	$\mathbb{Z}_2$	$\mathbb{Z}_6$	$\mathbb{Z}_3$

The $k=0$ column follows from the fact that $C_n(\mathbb{R}^2)$ is path-connected and the $k=1$ column can be obtained by abelianizing Artin’s presentation of $\mathbf{B}_n$. Even the low-degree calculations in Table 1 suggest a pattern: the homology of $\mathbf{B}_n$ in a fixed degree $k$ becomes independent of $n$ as $n$ increases.

Arnold proved the following stability result, in terms of the stabilization map $s_n\colon \mathbf{B}_n\hookrightarrow \mathbf{B}_{n+1}$ defined by adding an unbraided $(n+1)^{st}$ strand as in Figure 12.

Figure 12.

The stabilization map $s_3\colon \mathbf{B}_3\hookrightarrow \mathbf{B}_{4}$.

Theorem 2.3 (Arnold Arn70).

For each $k\geq 0$, the induced map

$$(s_n)_*\colon H_k(\mathbf{B}_n;\mathbb{Z})\rightarrow H_k(\mathbf{B}_{n+1};\mathbb{Z})$$

is an isomorphism for $n\geq 2k$.

The family $\{C_n(\mathbb{C})\}_n$ therefore satisfies homological stability. Arnold in fact proved the result for cohomology, and Theorem 2.3 follows from the universal coefficients theorem.

May and Segal proved that the stable braid group $\mathbf{B}_{\infty }$ has the same homology as the path component of the trivial loop in the double loop space $\Omega ^2 S^2$. Fuks calculated the cohomology of braid groups with coefficients in $\mathbb{F}_2$. F. Cohen and Vaĭns̆teĭn computed the cohomology ring with coefficients in $\mathbb{F}_p$ (for $p$ an odd prime), and described $H^k(\mathbf{B}_n; \mathbb{Z})$ in terms of the groups $H^{k-1}(\mathbf{B}_n; \mathbb{F}_p)$ ($p \text{ prime})$ for $k \geq 2$.

2.4. Homological stability for configuration spaces

For a $d$-manifold $M$, it is possible to visualize homology classes in $F_n(M)$ and $C_n(M)$ concretely. Consider Figure 13. This figure shows a $2$-parameter family of configurations in $F_n(M)$, in fact (because the two loops do not intersect) it shows an embedded torus $S^1 \times S^1 \hookrightarrow F_5(M)$. Thus, up to sign, this figure represents an element of $H_2(F_5(M))$. In a sense, the loop traced out by particle $3$ arises from the homology of the surface $M$, and the loop traced out by particle $4$ arises from the homology of $F_n(\mathbb{R}^d)$. From the homology of $M$ and $F_n(\mathbb{R}^d)$, it is possible to generate lots of examples of homology classes in $F_n(M)$. The problem of understanding additive relations among these classes, however, is subtle, and the groups $H_k(F_n(M);\mathbb{Z})$ are unknown in most cases.

Figure 13.

A class in $H_2(F_5(M))$.

When $M$ is (punctured) Euclidean space, the (co)homology groups of $F_n(M)$ were computed by Arnold and Cohen. However, even in the case that $M$ is a genus-$g$ surface, we currently do not know the Betti numbers $\beta _k = \mathrm{rank}(H_k(F_n(M);\mathbb{Z}))$. Recently Pagaria computed the asymptotic growth rate in $n$ of the Betti numbers in the case $M$ is a torus. In the case of unordered configuration spaces, in 2016 Drummond-Cole and Knudsen computed the Betti numbers of $C_n(M)$ for $M$ a surface of finite type.

Even though the (co)homology groups of configurations spaces remain largely mysterious, the tools of homological stability give us a different approach to understanding their structure.

Theorem Figure 2.3 on stability for braid groups raises the question of whether the unordered configurations spaces $\{C_n(M)\}_n$ satisfy homological stability for a larger class of topological spaces $M$. Let $M$ be a connected manifold. To generalize Theorem Figure 2.3 we must define stabilization maps

$$\begin{align*} C_n(M) & \longrightarrow C_{n+1}(M) \\ \{x_1, \ldots , x_n\} & \longmapsto \{x_1, \ldots , x_n, x_{n+1}\}. \end{align*}$$

Unfortunately, in general there is no way to choose a distinct particle $x_{n+1}$ continuously in the inputs $\{x_1, \ldots , x_n\}$, and no continuous map of this form exists. To define the stabilization maps, we must assume extra structure on $M$, for example, assume that $M$ is the interior of a manifold with nonempty boundary. Then, if we choose a boundary component, it is possible to define the stabilization map $s_n\colon C_n(M) \to C_{n+1}(M)$ by placing the new particle in a sufficiently small collar neighbourhood of the boundary component. This procedure (illustrated in Figure 14) is informally described as ‘adding a particle at infinity.’

Figure 14.

Stabilization map $s_3\colon C_3(M)\rightarrow C_4(M)$.

In the 1970s McDuff proved that the sequence $\{C_n(M)\}_n$ satisfies homological stability and Segal gave explicit stable ranges.

Theorem 2.4 (McDuff McD75; Segal Seg79).

Let $M$ be the interior of a compact connected manifold with nonempty boundary. For each $k \geq 0$ the maps

$$(s_n)_*\colon H_k(C_n(M);\mathbb{Z}) \longrightarrow H_k(C_{n+1}(M);\mathbb{Z})$$

are isomorphisms for $n \geq 2k$.

Concretely, this theorem states that degree-$k$ homology classes arise from subconfigurations on at most $2k$ particles. Heuristically, these homology classes have the form of Figure 15.

Figure 15.

A homology class after stabilizing by the addition of $n-2k$ particles.

Moreover, McDuff related the homology of the stable space $C_{\infty }(M)$ to the homology of $\Gamma (M)$, the space of compactly-supported smooth sections of the bundle over $M$ obtained by taking the fibrewise one-point compactification of the tangent bundle of $M$.

3. Other Stable Families

We briefly describe some other significant families satisfying (co)homological stability.

Symmetric groups. In Nak60 Nakaoka proved that the symmetric groups $\{S_n\}_n$ satisfy homological stability with respect to the inclusions $S_n\hookrightarrow S_{n+1}$. The Barratt–Priddy–Quillen theorem states that the infinite symmetric group $S_{\infty } = \bigcup _n S_n$ has the same homology of $\Omega _0^{\infty } S^{\infty }$, the path-component of the identity in the infinite loop space $\Omega ^{\infty } S^{\infty }$.

General linear groups. Let $R$ be a ring. Consider the sequence of general linear groups $\{\text{GL}_n(R)\}_n$ with the inclusions $\text{GL}_n(R)\hookrightarrow \text{GL}_{n+1}(R)$ given by

$$A\mapsto \begin{bmatrix} A & 0 \\ 0 & 1 \end{bmatrix}.$$

In the 1970s Quillen studied the homology of these groups when $R$ is a finite field $\mathbb{F}_q$ of characteristic $p$ in his seminal work on the $K$-theory of finite fields. He computes $H^*(\text{GL}_n(\mathbb{F}_q); \mathbb{F}_{\ell })$ for prime $\ell \neq p$ and determines a vanishing range for $\ell =p$.

In 1980 Charney proved homological stability when $R$ is a Dedekind domain. Van der Kallen, building on work of Maazen, proved the case that $R$ is an associative ring satisfying Bass’s “stable rank condition;” this arguably includes any naturally arising ring.

These results are part of a large stability literature on classical groups that warrants its own survey; see the extended version of this article for further references. Homological stability is known to hold for special linear groups, orthogonal groups, unitary groups, and other families of classical groups. There is ongoing work to study (co)homology with twisted coefficients, and sharpen the stable ranges.

Mapping class groups and moduli space of Riemann surfaces. Let $\Sigma _{g,1}$ be an oriented surface of genus $g$ with one boundary component and let the mapping class group

$$\text{Mod}(\Sigma _{g,1})\coloneq \pi _0(\text{Diff}^+(\Sigma _{g,1}\text{ rel }\partial ))$$

be the group of isotopy classes of diffeomorphisms of $\Sigma _{g,1}$ fixing a collar neighbourhood of the boundary. There is a map $t_g\colon \text{Mod}(\Sigma _{g,1})\hookrightarrow \text{Mod}(\Sigma _{g+1,1})$ induced by the inclusion $\Sigma _{g,1}\hookrightarrow \Sigma _{g+1,1}$ by extending a diffeomorphism by the identity on the complement $\Sigma _{g+1,1}\setminus \Sigma _{g,1}$, as in Figure 16.

Figure 16.

The map $\text{Mod}(\Sigma _{3,1})\to \text{Mod}(\Sigma _{4,1})$ is induced by the inclusion $\Sigma _{3,1}\hookrightarrow \Sigma _{4,1}$.

There is also a map $cap\colon \text{Mod}(\Sigma _{g,1})\rightarrow \text{Mod}(\Sigma _{g})$ induced by gluing a disk on the boundary component of $\Sigma _{g,1}$. Harer proved Har85 that the sequence $\{ \text{Mod}_{g,1}\}_{g}$ satisfies homological stability with respect to the inclusions $t_g$ and that for large $g$ the map $cap$ induces isomorphisms on homology. The proof and the stable ranges have been improved by the work of Ivanov, Boldsen, and others. Madsen and Weiss computed the stable homology by identifying the homology of mapping class groups, in the stable range, with the homology of a certain infinite loop space.

The rational homology of the mapping class group $\text{Mod}(\Sigma _g)$ is the same as that of the moduli space $\mathcal{M}_g$ of Riemann surfaces of genus $g\geq 2$. This moduli space parametrizes:

•

isometry classes of hyperbolic structures on $\Sigma _g$,

•

conformal classes of Riemannian metrics on $\Sigma _g$,

•

biholomorphism classes of complex structures on the surface $\Sigma _g$,

•

isomorphism classes of smooth algebraic curves homeomorphic to $\Sigma _g$.

One consequence of Harer’s stability theorem and the Madsen–Weiss theorem is their proof of Mumford’s conjecture: the rational cohomology of $\mathcal{M}_g$ is a polynomial algebra on generators $\kappa _i$ of degree $2i$, the so-called Mumford–Morita–Miller classes, in a stable range depending on $g$. See Tillman’s survey Til13.

Homological stability was established for mapping class groups of non-orientable surfaces by Wahl, for mapping class groups of some $3$-manifolds by Hatcher–Wahl and framed, Spin, and Pin mapping class groups by Randal-Williams.

Automorphism groups of free groups. Let $F_n$ denote the free group of rank $n$. Hatcher and Vogtmann proved that the sequence $\{ \text{Aut}(F_n)\}_n$ satisfies homological stability with respect to inclusions $\text{Aut}(F_n)\hookrightarrow \text{Aut}(F_{n+1})$. Galatius computed the stable homology by proving that $H_*(\text{Aut}(F_{\infty });\mathbb{Z})\cong H_*(\Omega _0^{\infty } S^{\infty };\mathbb{Z})\cong H_*(S_{\infty };\mathbb{Z})$. In particular, for