Geometry & Topology

There is a programme, largely developed by Weiss and Williams, that aims to understand the homotopy type of the diffeomorphism group of a compact high-dimensional manifold in terms of its algebraic K-theory (in the sense of Waldhausen). In this talk, I will give a brief overview of this programme and present an analogue for spaces of embeddings. The main difference is that, for embeddings, algebraic K-theory is often replaced by (relative) topological cyclic homology, a far more computable invariant.

I will also explain how, in joint work with João Lobo Fernandes, we use this analogous programme for embedding spaces to compute, in a range, the rational homotopy groups of diffeomorphism groups of many high-dimensional manifolds with infinite cyclic fundamental group, including solid tori $S^1 × D^n$ for $n > 4$, previously studied by Bustamante–Randal-Williams, Budney–Gabai, and Watanabe.