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 points in a fixed manifold, the Grassmannians of linear subspaces of dimension in and the moduli spaces , of Riemann surfaces of genus These families are usually indexed by some geometrically defined quantity, such as the number . of points in a configuration, the dimension of the linear subspaces, or the genus 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 we ask:
For many natural examples of moduli spaces some aspects of the topology get more complicated as the parameter , gets larger. For instance, the dimension of frequently increases with 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 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 of homotopy classes of loops in a topological space based at the point 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 . but they are often more computationally tractable and are widely studied. ,
Given a topological space and we can associate groups , and the , homology and cohomology groups (with coefficients in th where ), is a commutative ring such as or These algebraic invariants define functors from the category of topological spaces to the category of . for any continuous map of topological spaces -modules: there are induced maps -linear
The cohomology groups in fact have the structure of a graded with respect to the cup product operation. -algebra
The group is the free abelian group on the path components of the topological space and is its dual. If is path-connected, is naturally isomorphic to the abelianization of with respect to any basepoint and its elements are certain equivalence classes of (unbased) loops in ,.
For a topological group there exists an associated classifying space for principal . It is constructed as the quotient of a (weakly) contractible space -bundles by a proper free action of The space . is unique up to (weak) homotopy equivalence. If is a discrete group, then is precisely an Eilenberg-MacLane space , i.e., a path-connected topological space with and trivial higher homotopy groups. For example, up to homotopy equivalence, is the circle, is the infinite-dimensional real projective space and the Grassmanian of , linear subspaces in -dimensional is .
Some motivation to study the cohomology of its cohomology classes define characteristic classes of principal : invariants that measure the ‘twistedness’ of the bundle. For instance the cohomology algebra -bundles, can be described in terms of Pontryagin and Stiefel–Whitney classes.
With we can define the group homology and group cohomology of a discrete group by
We can refine Question 1.1 to the following:
In this article we discuss Question 1.2 with a particular focus on the families of configuration spaces and braid groups. For further readingFootnote1 we recommend R. Cohen’s survey Coh09 on stability of moduli spaces.
1.2. Homological stability
If the maps are inclusions we define to be the stable group or space. Under mild assumptions, if satisfies homological stability, then
We call the groups the stable homology.
2. An Example: Configuration Spaces and the Braid Groups
2.1. A primer on configuration spaces
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 is as the complement of the union of subspaces of defined by equations of the form .
In other words, we can construct by deleting the “fat diagonal” of consisting of all , in -tuples where two or more components coincide. In the simplest case, when and is the interval we see that , consists of two contractible components, as in Figure 1.
Another way we can conceptualize is as the space of embeddings of the discrete set into appropriately topologized. We may visualize a point in , by labelling points in as in Figure ,2.
From this perspective, we may reinterpret the path components of 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.
Any path through that interchanges the relative positions of the two particles must involve a ‘collision’ of particles, and hence exit the configuration space We encourage the reader to verify that, in general, the configuration space . is the union of contractible path components, indexed by elements of the symmetric group See Figure .4.
In contrast, if is a connected manifold of dimension or more, then is path-connected: given any two configurations, we can construct a path through from one configuration to the other without any ‘collisions’ of particles. In this case for all and this is our first glimpse of stability in these spaces as ,.
For any space the symmetric group , acts freely on by permuting the coordinates of an -tuple equivalently, by permuting the labels on a configuration as in Figure ,2. The orbit space is the (unordered) configuration space of particles on . This is the space of all subsets of -element topologized as the quotient of , The reader may verify that the quotient map (illustrated in Figure .5) is a regular space map. In particular, by covering space theory, the quotient map -covering induces an injective map on fundamental groups.
In the case that is the complex plane we can identify , with the space of monic degree- polynomials over with distinct roots, by mapping a configuration to the polynomial For this reason the topology of . has deep connections to classical problems about finding roots of polynomials.
We will address Question 1.2 for the families and but we first specialize to the case when , Although the spaces . and are path-connected, in contrast to the configuration spaces of 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 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 .
We can understand as follows. Choose a basepoint configuration in and then we may visualize a loop as a ‘movie’ where the , particles continuously move around 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 as the configuration space of the open 2-disk.
Loops in are similar, with the crucial distinction that the particles are unlabelled and indistinguishable, and so need only return set-wise to their basepoint configuration.
It is traditional to represent elements of the group and its subgroup by equivalence classes of braid diagrams, as illustrated in Figure 7. These braid diagrams depict strings (called strands) in Euclidean 3-space, anchored at their tops at distinguished points in a horizontal plane, and anchored at their bottoms at the same 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.
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 ,
uses generators corresponding to half-twists of adjacent strands, as in Figure 9.
Artin also gave a finite presentation for We will not state it in full, but comment that there are . generators ,( , corresponding to full twists of each pair of strands, as in Figure 10.
Corresponding to the regular covering space map of Figure 5, there is a short exact sequence of groups
The quotient map shown in Figure ,11, takes a braid, forgets the strands and simply records the permutation induced on their endpoints. The generator maps to the simple transposition The kernel is those braids that induce the trivial permutation, i.e., the pure braid group. .
2.3. Homological stability for the braid groups
Arnold calculated some homology groups of in low degree (Table 1).
The homology groups Empty spaces are zero groups. Stable groups are shaded. .
The column follows from the fact that is path-connected and the column can be obtained by abelianizing Artin’s presentation of Even the low-degree calculations in Table .1 suggest a pattern: the homology of in a fixed degree becomes independent of as increases.
Arnold proved the following stability result, in terms of the stabilization map defined by adding an unbraided strand as in Figure 12.
The family 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 has the same homology as the path component of the trivial loop in the double loop space Fuks calculated the cohomology of braid groups with coefficients in . F. Cohen and Vaĭns̆teĭn computed the cohomology ring with coefficients in . (for an odd prime), and described in terms of the groups ( for .
2.4. Homological stability for configuration spaces
For a -manifold it is possible to visualize homology classes in , and concretely. Consider Figure 13. This figure shows a family of configurations in -parameter in fact (because the two loops do not intersect) it shows an embedded torus , Thus, up to sign, this figure represents an element of . In a sense, the loop traced out by particle . arises from the homology of the surface and the loop traced out by particle , arises from the homology of From the homology of . and it is possible to generate lots of examples of homology classes in , The problem of understanding additive relations among these classes, however, is subtle, and the groups . are unknown in most cases.
When is (punctured) Euclidean space, the (co)homology groups of were computed by Arnold and Cohen. However, even in the case that is a genus- surface, we currently do not know the Betti numbers Recently Pagaria computed the asymptotic growth rate in . of the Betti numbers in the case is a torus. In the case of unordered configuration spaces, in 2016 Drummond-Cole and Knudsen computed the Betti numbers of for 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 satisfy homological stability for a larger class of topological spaces Let . be a connected manifold. To generalize Theorem Figure 2.3 we must define stabilization maps
Unfortunately, in general there is no way to choose a distinct particle continuously in the inputs and no continuous map of this form exists. To define the stabilization maps, we must assume extra structure on , for example, assume that , is the interior of a manifold with nonempty boundary. Then, if we choose a boundary component, it is possible to define the stabilization map 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.’
In the 1970s McDuff proved that the sequence satisfies homological stability and Segal gave explicit stable ranges.
Concretely, this theorem states that degree- homology classes arise from subconfigurations on at most particles. Heuristically, these homology classes have the form of Figure 15.
Moreover, McDuff related the homology of the stable space to the homology of the space of compactly-supported smooth sections of the bundle over , obtained by taking the fibrewise one-point compactification of the tangent bundle of .
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 satisfy homological stability with respect to the inclusions The Barratt–Priddy–Quillen theorem states that the infinite symmetric group . has the same homology of the path-component of the identity in the infinite loop space ,.
General linear groups. Let be a ring. Consider the sequence of general linear groups with the inclusions given by
In the 1970s Quillen studied the homology of these groups when is a finite field of characteristic in his seminal work on the of finite fields. He computes -theory for prime and determines a vanishing range for .
In 1980 Charney proved homological stability when is a Dedekind domain. Van der Kallen, building on work of Maazen, proved the case that 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 be an oriented surface of genus with one boundary component and let the mapping class group
be the group of isotopy classes of diffeomorphisms of fixing a collar neighbourhood of the boundary. There is a map induced by the inclusion by extending a diffeomorphism by the identity on the complement as in Figure ,16.
There is also a map induced by gluing a disk on the boundary component of Harer proved .Har85 that the sequence satisfies homological stability with respect to the inclusions and that for large the map 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 is the same as that of the moduli space of Riemann surfaces of genus This moduli space parametrizes: .
isometry classes of hyperbolic structures on ,
conformal classes of Riemannian metrics on ,
biholomorphism classes of complex structures on the surface ,
isomorphism classes of smooth algebraic curves homeomorphic to .
One consequence of Harer’s stability theorem and the Madsen–Weiss theorem is their proof of Mumford’s conjecture: the rational cohomology of is a polynomial algebra on generators of degree the so-called Mumford–Morita–Miller classes, in a stable range depending on , 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 by Hatcher–Wahl and framed, Spin, and Pin mapping class groups by Randal-Williams. -manifolds
Automorphism groups of free groups. Let denote the free group of rank Hatcher and Vogtmann proved that the sequence . satisfies homological stability with respect to inclusions Galatius computed the stable homology by proving that .