Eilenberg Lectures, Spring 2024
by Soren Galatius, monday 4:105:25PM in room 520.
Lecture overview
 Lecture 1, Jan 22
 Introduction. The many points of view on M_{g}: holomorphic (Riemann surfaces), algebraic geometric (complex algebraic curves), differential geometric (constant curvature), group theoretic (mapping class groups), homotopy theoretic (classifying space for diffeomorphism groups). The homotopy theoretic incarnation generalizes well to other dimensions. Statements of some theorems (Harer, Tillmann, MadsenWeiss; myself and RandalWilliams). See this survey for an overview of several papers with RandalWilliams.
 Lecture 2, Jan 29
 Cobordism categories as (nonunital) topological categories. To a topological category C we associate an ordinary category hC called the homotopy category, and a topological space BC called the classifying space.
 Lecture 3, Feb 5
 More on topological categories. The group completion map Hom_{C}(x,y) → Ω_{x,y}BC. The group completion theorem in the special case of topological monoids (and a promise that the proof by McDuff and Segal sometimes can be adapted to the multiobject case). Spectra, homotopy groups of spectra, weak equivalences between spectra, and the functor Ω^{∞} to pointed spaces. Thom spectra and the statement of the main result of this paper with Madsen, Tillmann, and Weiss. Tangential structures on ddimensional vector bundles and ddimensional manifolds.
 Lecture 4, Feb 12

Characteristic classes of smooth fiber bundles equipped with Θstructure: starting from a characteristic class of vector bundles with Θstructure, apply that to the fiberwise tangent bundle and then fiber integrate (these are the generalized MillerMoritaMumford classes see section 3.1 of the abovementioned survey).
There is a classifying space for smooth bundles equipped with Θstructure, and having defined the MillerMoritaMumford classes κ_{c} gives us a ring map from the free gradedcommutative algebra on abstract symbols κ_{c} to the classifying space for all smooth bundles with ddimensional closed fibers equipped with Θstructures. The group completion map from last time — from a Homspace in the cobordism category to a path space in the classifying space — gives a spacelevel instantation of this. In other words, rational cohomology of (any path component of) the codomain of the group completion map can be identified with the free gradedcommutative algebra on those symbols, and that the induced map on cohomology can be identified with evaluating a symbol κ_{c} on the universal bundle.
In the last part of the lecture, I then explained the main result asserting that the group completion map becomes acyclic after taking a suitable direct limit. I made an attempt to state this theorem in the generality in which we prove it, which becomes a bit technical. The statement is Theorem 1.5 and 7.3 of this paper with RandalWilliams.
The flavor of the general result is that one looks at a moduli space of null bordisms of some object of the cobordism category with Θstructure. Not all null bordisms, only those where the tangential structure is an nconnected map (I spelled out what that means in two cases: d=2 and Θ = {*} as well as d=4 and Θ = {± 1}). The group completion map restricts to a map from this space of null bordisms to a space of paths in BC_{d,Θ}, and the main result is that this restricted group completion map becomes an acyclic map after taking a suitable colimit over a sequence of maps induced by morphisms in the cobordism category. This sequence of cobordisms should satisfy some assumptions, including that each of them should contain S^{n} x S^{n} as a connected summand. In that sense, we "take connected sum with infinitely many copies of S^{n} x S^{n}".
 Lecture 5, Feb 19
 Brief discussion of the difference between acyclic maps and weak homotopy equivalences (for example, if X is the space obtained from a homology sphere by removing a point, and Y is a onepoint space, then the unique map X → Y is acyclic). Brief discussion of Postnikov truncation (see for instance Example 4.16 on page 354 of Hatcher's book, the idea is that τ_{<n} X is obtained from X by attaching (n+1)cells along all possible maps in order to kill π_{n}, followed by attaching (n+1)cells in all possible ways, and so on).
The main theorem from last time in particular gives a formula for H_{0}, degreezero homology, in the direct limit over taking connected sum with more and more copies of S^{n} x S^{n}. This can be interpreted as a classification result for manifolds (with tangential structure) up to connected sum with S^{n} x S^{n}, in the spirit of Wall and Kreck (Theorem C of this paper). The main result stated last time can therefore be viewed as generalizations of those results to higher homological degrees.
The notions of orientation on 2manifolds and spin structures on 4manifolds are in fact the first two steps in a sequence of canonical tangential structures on evendimensional manifolds — the corresponding notion on 2ndimensional manifolds has Θ = τ_{<n} GL_{2n}(R) for any n. This tangential structure plays a special role in the theory, but we also discussed that for the special manifolds W_{g,1} = D^{2n}#g(S^{n} x S^{n}) the space of structures is contractible (more precisely, the space of structures extending a fixed structure near on the boundary). Therefore, the space of highly connected null bordisms equipped with a structure can be identified with just a moduli space of certain manifolds without any tangential structure. The structure Θ = τ_{<n} GL_{2n}(R) is nevertheless important for this example, since it appears in the formula for what the limiting homology and cohomology is.