Eilenberg Lectures, Spring 2024

by Soren Galatius, monday 4:10-5:25PM in room 520.

Lecture overview

Lecture 1, Jan 22
Introduction. The many points of view on Mg: 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, Madsen-Weiss; myself and Randal-Williams). See this survey for an overview of several papers with Randal-Williams.
Lecture 2, Jan 29
Cobordism categories as (non-unital) 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 HomC(x,y) → Ωx,yBC. 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 multi-object 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 d-dimensional vector bundles and d-dimensional manifolds.
Lecture 4, Feb 12
Characteristic classes of smooth fiber bundles equipped with Θ-structure: starting from a characteristic class of vector bundles with Θ-structure, apply that to the fiberwise tangent bundle and then fiber integrate (these are the generalized Miller-Morita-Mumford classes see section 3.1 of the above-mentioned survey).
There is a classifying space for smooth bundles equipped with Θ-structure, and having defined the Miller-Morita-Mumford classes κc gives us a ring map from the free graded-commutative algebra on abstract symbols κc to the classifying space for all smooth bundles with d-dimensional closed fibers equipped with Θ-structures. The group completion map from last time — from a Hom-space in the cobordism category to a path space in the classifying space — gives a space-level 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 graded-commutative algebra on those symbols, and that the induced map on cohomology can be identified with evaluating a symbol κc 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 Randal-Williams.
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 Θ-structure. Not all null bordisms, only those where the tangential structure is an n-connected 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 BCd,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 Sn x Sn as a connected summand. In that sense, we "take connected sum with infinitely many copies of Sn x Sn".
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 one-point 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 πn, 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 H0, degree-zero homology, in the direct limit over taking connected sum with more and more copies of Sn x Sn. This can be interpreted as a classification result for manifolds (with tangential structure) up to connected sum with Sn x Sn, 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 2-manifolds and spin structures on 4-manifolds are in fact the first two steps in a sequence of canonical tangential structures on even-dimensional manifolds — the corresponding notion on 2n-dimensional manifolds has Θ = τ<n GL2n(R) for any n. This tangential structure plays a special role in the theory, but we also discussed that for the special manifolds Wg,1 = D2n#g(Sn x Sn) 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 GL2n(R) is nevertheless important for this example, since it appears in the formula for what the limiting homology and cohomology is.