The homotopy type of the PL cobordism category. II

TL;DR

This paper establishes the homotopy type of the PL cobordism category by proving a PL analogue of a key theorem, introducing a new spectrum, and demonstrating a weak homotopy equivalence.

Contribution

It introduces the PL Madsen-Tillmann spectrum and proves the homotopy equivalence of the PL cobordism category with an infinite loop space.

Findings

Proves the PL analogue of the Galatius-Madsen-Tillmann-Weiss theorem.

Defines the PL Madsen-Tillmann spectrum $ extbf{MT}PL(d)$.

Establishes a weak homotopy equivalence $B ext{Cob}^{PL}_d o \

Abstract

In this article, we prove the PL analogue of the theorem of Galatius, Madsen, Tillmann, and Weiss which describes the homotopy type of the smooth cobordism category. More specifically, we introduce the PL Madsen-Tillmann spectrum $\mathbf{MT}PL(d)$ and prove that there is a weak homotopy equivalence of the form $B\mathsf{Cob}^{\mathrm{PL}}_d \simeq \Omega^{\infty-1}\mathbf{MT}PL(d)$. We also discuss how to adjust the methods of this paper to obtain the topological version of our main result.