Algebra/Topology Seminar 07122020

Speaker: Jan Steinebrunner

Title: The reduced one-dimensional bordism category and TC of simply connected spaces

Abstract: The topologically enriched bordism category Bord_d has as objects closed oriented (d-1)-manifolds and as morphism spaces the moduli spaces of oriented d-bordisms. The classifying space B(Bord_d) was computed by Galatius-Madsen-Tillmann-Weiss, and has been used to great success in the study of moduli spaces.

In this talk, after recalling Bord_d, I will focus on its much simpler predecessor: the homotopy category h(Bord_d) where any two diffeomorphic bordisms are identified. Surprisingly little is known about the homotopy type of h(Bord_d). I will explain how to compute the classifying space of h(Bord_1) in terms of CP^\infty_{-1} = MTSO_2. The proof makes use of a new 'reduced' bordism category Bord_1^{red} where all circles are deleted. More generally, one can consider Bord_1^{red}(X) where all bordisms are labelled by a space X. When X is simply connected, this turns out to be closely related to the topological cyclic homology TC(S[\Omega X]).

If time permits, I might also show how to construct cocycles for an infinite family of non-trivial cohomology classes kappa_i on h(Bord_1), and how this can be used to show that a large subcategory of h(Bord_2) is highly non-trivial.