The homotopy type of the PL cobordism category. I

TL;DR

This paper defines a PL cobordism category and proves its classifying space is weak homotopy equivalent to an infinite loop space, advancing the understanding of PL topology analogous to smooth cobordism theories.

Contribution

It introduces a PL cobordism category and establishes its classifying space's homotopy type, bridging PL topology with infinite loop space theory.

Findings

Classifying space of PL cobordism category is weak homotopy equivalent to an infinite loop space.

Provides a foundation for Madsen-Weiss type results in PL topology.

Extends smooth cobordism concepts to the piecewise linear setting.

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.