The classical approach to studying the enriched category of manifolds and their diffeomorphisms is to first compare it to a simplified category of block-manifolds and then compare the latter to the category of Poincaré complexes. The information lost in each of these steps is encoded in certain structure spaces that are expressible—in a certain range—in terms of K- and L-theory.
More recent developments related to manifold calculus and factorisation homology suggest a different approach, namely to compare the category of manifolds to a variant of the derived category of modules over the little d-discs operad. Again, this amounts to studying certain structure spaces that encode the difference: the Disc-structure spaces.
In this talk, I will explain the above and describe aspects of joint work with A. Kupers in which we show that, in most cases, these Disc-structure spaces are nontrivial infinite loop spaces that depend only little on the underlying manifolds.