The classifying space of the one-dimensional bordism category and a cobordism model for TC of spaces

Abstract

The homotopy category of the bordism category $hBor{d}_d$ has as objects closed oriented $(d-1)$-manifolds and as morphisms diffeomorphism classes of $d$-dimensional bordisms. Using a new fiber sequence for bordism categories, we compute the classifying space of $hBor{d}_d$ for $d=1$, exhibiting it as a circle bundle over ${\mathbb{CP}}_{-1}^{\infty }$. As part of our proof we construct a quotient $Bor{d}_1^{red}$ of the cobordism category where circles are deleted. We show that this category has classifying space ${\Omega }^{\infty -2}{\mathbb{CP}}_{-1}^{\infty }$ and moreover that, if one equips these bordisms with a map to a simply connected space $X$, the resulting $Bor{d}_1^{red}(X)$ can be thought of as a cobordism model for the topological cyclic homology $TC(\mathbb{S}[\Omega X])$. In the second part of the paper we construct an infinite loop space map $B(hBor{d}_1^{red})\to Q({\Sigma }^2{\mathbb{CP}}_{+}^{\infty })$ in this model and use it to derive combinatorial formulas for rational cocycles on $Bor{d}_1^{red}$ representing the Miller-Morita-Mumford classes ${\kappa }_i\in H^{2i+2}((B(hBor{d}_1);\mathbb{Q})$.

46 pages, 9 figures. Final version, to appear in the Journal of Topology