EPFL Topology Seminar Spring 2024

Traces of dualizable categories and functoriality of the Becker-Gottlieb transfers

For any fiber bundle with compact smooth manifold fiber X ⟶ Y, Becker and Gottlieb have defined a “wrong way” map S[Y] ⟶ S[X] at the level of homology with coefficients in the sphere spectrum. Later on, these wrong way maps have been defined more generally for continuous functions whose homotopy fibers are finitely dominated, and have been since referred to as the Becker-Gottlieb transfers. It has been a long standing open question whether these transfers behave well under composition, i.e. if they can be used to equip homology with a contravariant functoriality.

In this talk, we will approach the transfers from the perspective of sheaf theory. We will recall the notion of a locally contractible geometric morphism, and then define a Becker-Gottlieb transfer associated to any proper, locally contractible map between locally contractible and locally compact Hausdorff spaces. We will then use techniques coming from recent work of Efimov on localizing invariants and dualizable stable infinity-categories to construct fully functorial “categorified transfers”. Functoriality of the Becker-Gottlieb transfers is then obtained by applying topological Hochschild homology to the categorified transfers.

If time permits, we will also explain how one can use similar methods to extend the Dwyer-Weiss-Williams index theorem for compact topological manifolds fiber bundles to proper locally contractible maps. In particular, this shows that the homotopy fibers of a proper locally contractible map are homotopy equivalent to finite CW-complexes. Therefore, it is still unclear whether functoriality of the transfers associated to maps with finitely dominated homotopy fibers holds.

This is a joint work with Maxime Ramzi and Sebastian Wolf.

Benjamin Antieau

Perfection and homotopy type

I will discuss the role of perfection in classifying homotopy types by their integral cohomology, building on work of Sullivan, Kriz, Mandell, and Yuan. The main idea involves binomial rings, which in the p-complete case are delta-rings for which the associated Frobenius endomorphism is the identity. This idea has been discovered independently by Horel and by Kubrak—Shuklin—Zakharov.

Let C be a stable infinity-category equipped with a bounded t-structure with the heart denoted by A. Antieau, Gepner, and Heller conjectured that the map of nonconnective K-theory spectra K(A) —-> K(C) is always an equivalence. Barwick’s theorem of the heart implies that this map is an equivalence on connective covers, and both sides are known to be connective if A is a noetherian abelian category.

To any spectrum M we functorially assign a stable infinity-category C_M such that the spectrum K(C_M) is equivalent to M.

Using this result and some basic chromatic homotopy theory, we construct a counterexample to the conjecture above.

Models for configuration spaces via obstruction theory

For M a parallelized n-manifold, the configuration spaces FM_M(r) can be given the structure of a right E_n-module. Through rational homotopy theory, this then gives rise to a right comodule over the Hopf cooperad H^\bullet(E_n). I will show how to classify all H^\bullet(E_n)-comodules of “configuration space type”.

A simple proof of the Mumford conjecture

Andrea Bianchi recently gave a new proof of Mumford’s conjecture on the stable rational cohomology of the moduli space of curves (first proven by Madsen and Weiss). I will explain a streamlined and simplified version of Bianchi’s argument. (Joint with Ronno Das)