Algebra/Topology Seminar

Title: A model for the assembly map of bordism-invariant functors

Speaker: Guglielmo Nocera

Abstract: Let G be a discrete group. The Borel Conjecture asserts that for every pair of closed topological manifolds M, N both homotopy equivalent to BG, then M and N are homeomorphic.

Known cases of the Borel conjecture have been proven either via classification of low-dimensional manifolds or as a consequence of known cases of the so-called Farrell-Jones Conjecture: this is an algebraic criterion involving the notions of K- and L-theory of manifolds with coefficients in a ring R and asserting that certain assembly maps are equivalences of spectra.

There is an ongoing program of revisiting the Farrell-Jones conjecture by means of ∞-categorical tools, mainly Efimov’s continuous K-theory on one hand and hermitian K-theory on the other, with the goal of potentially proving unknown cases. For instance, continuous K-theory offers a categorical model for the assembly map in K-theory.

In joint work with Jordan Levin and Victor Saunier, we use hermitian K-theory to offer a categorical model of the assembly map for L-theory, i.e. a functor between Poincar´e categories whose L-theory is precisely the L-theoretic assembly map. This categorical model is indeed a Verdier projection, whose kernel we explicitly describe.

As a direct application, we generalize the Shaneson splitting for bordism-invariant functors of Poincar´e categories proved by Calm`es–Dotto–Harpaz–Hebestreit–Land–Moi–Nardin–Nikolaus–Steimle to allow for twists. We thus recover Ranicki’s exact sequence theorem for the L-theory of twisted Laurent extensions.