The homotopy type of the PL cobordism category. I

Abstract

In this paper, we introduce a bordism category $\mathcal{C}_d^{PL}$ whose objects are bundles of closed $(d-1)$-dimensional piecewise linear manifolds and whose morphisms are bundles of $d$-dimensional piecewise linear cobordisms. In the main theorem of this article, we show that the classifying space $B\mathcal{C}_d^{PL}$ is weak homotopy equivalent to an infinite loop space. We regard $\mathcal{C}_d^{PL}$ as the piecewise linear analogue of the category of smooth cobordisms which has been studied extensively in connection to the Madsen-Weiss Theorem, and the main result of this paper is a first step towards obtaining Madsen-Weiss type results in the context of piecewise linear topology.