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 $hBord_d$ 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 $hBord_d$ for $d = 1$, exhibiting it as a circle bundle over $\mathbb{CP}^\infty_{-1}$. As part of our proof we construct a quotient $Bord_1^{red}$ of the cobordism category where circles are deleted. We show that this category has classifying space $\Omega^{\infty-2}\mathbb{CP}^\infty_{-1}$ and moreover that, if one equips these bordisms with a map to a simply connected space $X$, the resulting $Bord_1^{red}(X)$ 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 $B(hBord_1^{red}) \to Q(\Sigma^2 \mathbb{CP}^\infty_+)$ in this model and use it to derive combinatorial formulas for rational cocycles on $Bord_1^{red}$ representing the Miller-Morita-Mumford classes $\kappa_i \in H^{2i+2}((B(hBord_1); \mathbb{Q})$.