The goal of this talk is to understand the “moduli space of -dimensional (compact) manifolds”. Of course, -dimensional manifolds are not very interesting. They are just a finite collection of points. The diffeomorphism classes of -dimensional manifolds are indexed by the natural numbers , and the automorphism group of is . So the (homotopy type of) the moduli space is
The disjoint union of manifolds endows with the structure of an -monoid. This is the correct notion of a topological monoid in the homotopical setting. It is in fact , but we shall focus on the structure. We will provide a concrete definition of an -monoid later on and explicitly describe this structure.
If is any -monoid, then is a monoid, which may or may not be a group. We say is group-like (or an -group) if is a group, and the group completion of , written , is the universal map of -monoids from to an -group. In the previous talk, we discussed the group completion theorem, which relates the homology of to that of .
There is an equivalence of -groups
There is an “easy” abstract nonsense proof of this theorem — simply observe that both sides are the free -group on a point. However, the perspective we want to take is that is the moduli of -dimensional manifolds, and so we want a proof that adopts this perspective.
In this talk, we will construct an explicit geometric model of and endow it with the structure of an -monoid. We will then use this explicit model to prove the equivalence above.
First of all, instead of thinking about “all” -dimensional manifolds, we consider -dimensional submanifolds of . In general,
If is a manifold, write for the space of distinct (unordered) points on .
It is not difficult to see that this is a model of , using the fact that the space of embeddings of points in is connected. So we have
It will be useful to consider a variant of this configuration space:
is the set of discrete subsets of , where a sequence of configurations converges if the intersection with any open ball converges.
We shall define two contractible open subsets such that . Then we have a homotopy pushout square
that exhibits as the suspension of , hence is .
We define the subsets as follows:
contains the configurations with no point at the origin.
contains the configurations with a unique point closest to the origin.
These indeed cover — if a configuration is not in , then it contains a point at the origin, which is necessarily the unique point closest to the origin. We observe
deformation retracts to by pushing every point away from the origin;
deformation retracts to by translating the distinguished point to the origin, then pushing all other points out to infinity;
deformation retracts onto by scaling the distinguished point to radius , then pushing all other points to infinity.
We generalize the definition of a bit:
If is open, we define to be the subset of consisting of the configurations that are contained in .
If is the interval , then
The main theorem we have to prove is the following:
Let be an open subspace of the form with precompact. Then has a canonical structure as an -monoid and
Iterating this with the case , we find that
The idea of the theorem is that if were a topological monoid, then the theorem is equivalent to , and in the bar construction for , we have lots of copies of put next to each other, which give us an .
To actually prove the theorem, we need to first know what it means to be an -monoid. It turns out the definition of an -monoid is one such that the idea above can be made literally true.
Our notion of an -monoid is what people call a reduced Segal space. The idea is that a reduced Segal space is a simplicial space that “looks like” the bar resolution of a topological monoid.
An -monoid is a (proper) simplicial space such that the maps given by the inclusions sending is a weak equivalence. In particular, .
Given an -monoid , we will refer to as the underlying space, and we expect the geometric realization to be the delooping of . The main theorem (whose proof we omit) is the following:
If is an -monoid, then .□
So to prove the main theorem, we need to construct such that and .
We define to be the subset of consisting of elements such that never meets the “walls” . The face map forgets the wall , and the degeneracies repeat the corresponding wall. 1
There is an inclusion , which is a deformation retract, and in particular a homotopy equivalence. This deformation retract is simply given by scaling the the configuration and linearly until , and then pushing the points with away to infinity. Similarly, we get and we see that is indeed an -monoid.
We next need to show that . A point in is an element together with some non-zero weights on the summing to , modulo some equivalence relations. We define a map that simply forgets the walls and weights.
We first prove that this is a weak equivalence if is precompact. The strategy is to show that this map is a Serre microfibration with weakly contractible fibers. Recall that a Serre fibration is a map where we can always solve the lifting problem
A Serre microfibration is a weaker notion where we only need to be able to lift of the restriction of the bottom map to for some small . It is a theorem that a Serre microfibration with weakly contractible fibers is a weak equivalence.
The map is easily seen to be a microfibration in the case where is precompact, because any element in only has finitely many points and so the configuration of points is bounded away from the walls, so any small perturbation of the points will still not hit the walls. If is not precompact, this argument fails because we can have infinitely many points that can get arbitrarily close to the walls, and we need to modify our argument.
To see that the fibers are contractible, fix . Then is the space of ways to insert walls and weights that do not hit . If is a compact space and is a map, then by compactness, the walls in are bounded. So we can pick a so large that the walls in are all , and we may also assume does not hit . There is then a homotopy from to the constant map on by scaling down the weights at the existing walls and putting the weights on . So is contractible.
In the case where is not precompact, we have to do a bit more work. We define another space whose -simplices is the subspace of consisting of such that is disjoint from the subset of the wall (and we require if ). This is a less restrictive condition, and we have maps
The horizontal map sets and is a deformation retract because we can push points on away from the point (and then translate to ). Then the above argument generalizes to show that is a Serre microfibration with weakly contractible fibers.
This proof is meant to serve as a blueprint for understanding the moduli space of higher dimensional manifolds. In the positive dimensional case, essentially the same proof will show that (the classifying space of) the “cobordism category” is homotopy equivalent to an explicit infinite loop space. The main differences are as follows:
We have to relate this cobordism category to the moduli space of manifolds. In the -dimensional case, this requires almost no work. Indeed, we managed to avoid mentioning the cobordism category at all.
We have to identify the replacement of with an appropriate Thom space. This will be done with a technique known as scanning.
One has to put some effort into defining the right topology on the moduli spaces, which is more complicated than the zero-dimensional case. In the -dimensional case, we had this trick of considering the space in addition to , and in positive dimensions, we need to play multiple similar tricks to relate different models of the moduli space.