Online Talk

The classical approach to studying the (enriched) category of manifolds and diffeomorphisms is to compare it to the category of spaces and homotopy equivalences, by remembering the underlying homotopy type of a manifold. The difference between these categories is encoded in certain structure spaces that are long known to be expressible—in a range and up to extensions—in terms of algebraic K- and L-theory. More recent developments related to manifold calculus and factorisation homology suggest a different comparison, which in addition to the underlying homotopy type of a manifold also takes the homotopy types of all its configuration spaces into account. Joint work with A. Kupers shows, roughly speaking, that the latter captures a surprising amount of the category manifolds, in that the structure spaces encoding the difference—the eponymous Disc-structure spaces—have strong structural properties not shared by their classical analogues. In this talk, I will make the above precise and summarise what is currently known about these Disc-structures spaces.