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 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, 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 Hom_C(x,y) → Ω_x,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 κ_c gives us a ring map from the free graded-commutative algebra on abstract symbols κ_c 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 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 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 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 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 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 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 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 = 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 GL_2n(R) is nevertheless important for this example, since it appears in the formula for what the limiting homology and cohomology is.

Lecture 6, Feb 26

Homological stability, proved by Harer for oriented 2-manifolds, non-orientable 2-manifolds by Wahl, by Galatius and Randal-Williams for simply connected manifolds of any even dimension 2n≥6. The result with Randal-Williams was established in two papers, one about homological stability with respect to gluing an extra SⁿxSⁿ-summand, another about homological stability with respect to gluing k-handles for k≥n. Both have assumptions, roughly speaking that the underlying manifolds be simply connected and that they start out containing many SⁿxSⁿ-summands already (see the papers for precise statements, especially Corollaries 1.7 and 1.8 of the second paper).
In the last 20 minutes, I gave a brief introduction to the work of Kupers and Randal-Williams. This concerns the manifolds W_g,1 = D²ⁿ#g(SⁿxSⁿ), and for notational convenience we stuck to the case 2n=6 in this lecture. We defined the Torelli group as the subgroup of the diffeomorphism group (rel boundary) acting trivially on homology of the manifold, and discussed the Serre spectral sequence associated to the fiber sequence BTor(W_g,1) → BTor(W_g,1) → BSp_2g(Z). If one knew the cohomology of BTor(W_g,1) as a representation of the symplectic group, then one could write down the E₂-page of this spectral sequence and attempt to use it to calculate the cohomology of BDiff(W_g,1) in a different way. This is, very roughly, what Kupers and Randal-Williams do, except they use the argument in reverse to determine the cohomology of BTor(W_g,1) as a representation.
At the end of the lecture, I stated a theorem of Kupers that the cohomology of the Torelli group is finite dimensional in each cohomological degree (he proved more than that in his paper), and a theorem of Kupers and Randal-Williams that the cohomology of the Torelli group forms an algebraic representation of the symplectic group.

Lecture 7, Mar 4

Overview of the two papers of Kupers and Randal-Williams: The cohomology of Torelli groups is algebraic and On the cohomology of Torelli groups. They write W_g = g(Sⁿxⁿ) for the g-fold connected sum and define the Torelli group as the subgroup of Diff(W_g,D²ⁿ) consisting of diffeomorphisms acting trivially on H_n(W_g). The cohomology of (the classifying space of) these Torelli groups then come with an action of an arithmetic group. The goal of the two papers is to determine the rational cohomology of these Torelli groups in a stable range, as a representation.
I recalled what an arithmetic group is and what an algebraic representation of one is. The main result of the first of the papers by Kupers and Randal-Williams is that cohomology (in a fixed degree) of Torelli groups gives an algebraic representation, at least for 2n≥6. The second paper determines which representation it is.

Mar 11

Spring break, no lecture

Lecture 8, Mar 18

Strategy of the papers by Kupers and Randal-Williams, for calculating the cohomology of Torelli groups. The action of the diffeomorphism group of W_g on H = H_n(W_g;Q) = Q^2g gives a local system on BDiff(W_g,D) which is also denoted H. The tensor powers H^⊗q form local systems with action of the symmetric group S_q and in fact local systems of modules over the so-called Brauer algebra.
I sketched the ingredients involved in passing from knowledge of the cohomology groups H^*(BDiff(W_g,D);V^⊗q) in a range, as modules over the Brauer algebra, to knowing H^*(BTor(W_g,D)) as a representation of G_g' in a range. A main ingredient is Schur-Weyl duality, which for GL asserts that knowing a Q[S_q]-module V is equivalent to knowing the GL(W)-representation (V⊗W)^S_q for large enough vector spaces W. For the orthogonal group it asserts that knowing an O_2g-representation W is equivalent to knowing (W⊗H^⊗q)^O_2g as a module over the Brauer algebra Br_2g for all g.

Lecture 9, Mar 25

We first discussed another motivation for studying cohomology of Torelli groups, namely (for n≥3) that their classifying spaces are nilpotent. This is also a theorem of Kupers and Randal-Williams (Theorem C of this paper). The well-known Hurewicz theorem, asserting that a map of simply connected spaces is a weak homotopy equivalence if it induces an isomorphism in integral homology, holds more generally for nilpotent spaces. For nilpotent spaces one can also, at least in principle, hope to access their (rational) homotopy groups starting from their (rational) cohomology ring.
We then discussed the special role played by the tangential structures Θ^W = K(W,n+2) x τ_<nGL_2n(R), where W is a rational vector space and K(W,n+2) is the Eilenberg-MacLane space, equipped with trivial action of GL_2n(R). Applications to determining H^*(BDiff(W_g,D);H^⊗q) in a range, as a representation of the symmetric group S_q: this is accessed via determining the functor W ↦ H^*(BDiff(W_g,D);H^⊗q⊗W^⊗q)^S_q, from vector spaces to vector spaces. Using that the answer gives a free module over cohomology of the arithmetic group G_g', an elementary lemma about spectral sequences (see Lemma 4.3 of this paper) implies that differentials in the spectral sequence vanish in a range of total degrees. In this range of degrees, we can also deduce an isomorphism of S_q-representations (H^*(BTor(W_g,D);Q)⊗H^⊗q)^G'_g ≅ H^*(BDiff(W_g,D);H^⊗q)⊗_H^*(BG'g)Q.

Lecture 10, Apr 1

The conclusion from last lecture is a (complicated but explicit) formula for the Q[S_q]-module of maps of G'_g-representations H^⊗q→H^t(BTor(W_g,D);Q) for g≫t. In practice, since any algebraic representation of G'_g is a quotient of H^⊗q for some q, this allows a determination of (the algebraic part of) the G'_g-representation H^t(BTor(W_g,D);Q). The abstract reason for this is again Schur-Weyl duality (using that the S_q-equivariance is easily upgraded to linearity over the Brauer algebra). Time permitting, a discussion of the setup of Sam and Snowden for asymptotics of compatible sequences of representations of orthogonal and symplectic groups.

Apr 8

No lecture

Lecture 11, Apr 15

From H^*(BTor(W_g,D);Q) to more fundamental objects in geometric topology: BDiff_∂(D²ⁿ) and BTop(2n). These are related via smoothing theory, which gives an isomorphism of homotopy groups π_k+d(BDiff_∂(D²ⁿ)) = π_k(Top(d),O(d)). Surprisingly, it seems easier to access BDiff_∂(D²ⁿ) than Top(d) directly. The Weiss fiber sequence (section 4 of Kupers' paper) puts the homotopy groups of BDiff_∂(D²ⁿ) in a long exact sequence with the homotopy groups of the diffeomorphism group of W_g,1 and of a certain monoid of self-embeddings of W_g,1. In the next lecture we will discuss how this space of self-embeddings may be approached using embedding calculus.

Lecture 12, Apr 22

No formal lecture today, but I had "office hours" in the usual room.
We discussed the overall strategy of the paper On diffeomorphisms of even-dimensional discs by Kupers and Randal-Williams, and the role played by cohomology of Torelli groups. The point is that for a nilpotent space X with base point x, the rationalized homotopy groups π_*(X)⊗Q can be identified with the primitives in the the Hopf algebra H_*(Ω X)⊗Q, and may be calculated by a spectral sequence starting with the primitives in the Hopf algebra Ext_H^*(X;Q)(Q,Q), also known as the Harrison homology of the ring H^*(X;Q). This can be applied to the classifying spaces for the Torelli groups, to compute their rational homotopy groups in a range of degrees, as a representation.
To combine this with the Weiss fiber sequence from last week, we need a way of calculating rational homotopy groups of self-embedding spaces. This is the embedding calculus of Weiss and Goodwillie-Weiss. I emphasized that the main feature of this theory is that it presents (self-)embedding spaces Emb(M,N) as an inverse limit of approximations T_kEmb(M,N) and that the relative homotopy groups of two adjacent stages in the approximation has a useful description as the homotopy groups of a space of sections. This is carried out in section 5 of the paper of Kupers and Randal-Williams.

Lecture 13, Apr 29