Manifolds, homotopy theory, and related topics

Given a smooth projective complex variety X, one can wish to compute the homology of the space of holomorphic maps Hol(X,P^m) to projective space. A fruitful idea of Segal is to compare it to the space of continuous maps: one shows the inclusion is homology connected in a range of degrees. Going beyond that stable range is hard, but I will explain how to organise the problem using Weiss calculus applied to the unitary functors V -> \Sigma^\infty_+ Hol(X, P(V)) and \Sigma^\infty_+ Map(X, P(V)). I shall describe the two key points: the geometry of the space of holomorphic maps in terms of Stiefel manifolds, and how to use Goodwillie calculus in two variables.

The Higman-Thompson groups are groups of certain self-homeomorphisms of Cantor sets, which can be described using tree diagrams. We build a topological model for these groups where objects are configurations of points and morphisms are paths of configurations which are allowed to collide in certain ways. We use this model to compute the group homology in a stable range by constructing a scanning map following the work of Madsen, Weiss, Galatius and others. This map allows us to express the stable homology of the groups as the homology of the infinite loop space of a certain spectrum, and thereby recover a result of Szymik and Wahl.

Poincaré duality spaces - spaces having the homological properties of closed manifolds - are interesting subjects of study in geometric topology, in particular due to their relevance in classification problems. It is often useful to consider variants: with boundary, with an embedded subspace, equivariant and isovariant analogs. In my talk, I will present a unifying approach that covers many such variants, and demonstrate the main results of our theory in a few concrete examples. This is based on joint work with Bianchi, Hilman and Kirstein.

We define a cobordism category Cob_d^G for Poincaré complexes. It is an infinity-category with objects Poincaré complexes of dimension (d-1) and mapping spaces are given by classifying spaces of self-homotopy equivalences of d-dimensional Poincaré cobordisms. In this talk, we restrict to the oriented two-dimensional case Cob_2^SG. We can actually define a category Cob_2^SG(X) for any space X, by equipping objects and morphisms with a map to X. 

In particular, I will describe a formula for the classifying space BCob_2^SG(X). More importantly, I will use it to compute the first Goodwillie derivative of the functor X —> BCob_2^SG(X) as the Thom spectrum of a certain spherical fibration.

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