Manifolds and functor calculus

forthcoming

The study of smooth fibre bundles of compact high-dimensional manifolds connects several areas of mathematics such as algebraic and geometric topology, differential geometry, and number theory. From the perspective of homotopy theory, it amounts to the study of the homotopy type of the moduli space of manifolds. The two projects this proposal investigate this and related moduli spaces in using techniques from homotopy theory, especially different flavours of functor calculus. Classically, the moduli space of manifolds was related to homotopy-theoretic objects by considering a forgetful map to a moduli space of homotopy types and analysing the difference. Recent work of Kupers and I considered a moduli space of more refined homotopy-theoretic objects---which does not only remember the underlying homotopy type of a manifold, but also of all its configuration spaces---and showed that this homotopy theoretic moduli space is likely to be very close to the moduli space of manifolds. The first project in this proposal systematically studies these two moduli spaces and their difference, which will lead to deep new results both on the manifold as well as the homotopy-theoretic side. As an example, one of the objectives is to show that the passage from manifolds to the homotopy-theoretic objects is closely related to the cyclotomic trace from algebraic K-theory to topological cyclic homology. The second project develops a new approach to the relation between the moduli space of manifolds and algebraic K- and L-Theory using methods from parametrised surgery theory, homotopy calculus, and Grothendieck--Witt theory. In addition to providing simultaneously a new proof of the pseudoisotopy-stability theorem and the stable parametrised h-cobordism theorem, this approach has the potential to go far beyond what was known from the traditional approach and in particular answer long-standing questions on the pseudoisotopy stable range.