The homotopy type of the PL cobordism category, II

Gómez López, Mauricio

doi:10.2140/agt.2025.25.2613

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The homotopy type of the PL cobordism category, II

Mauricio Gómez López

Algebraic & Geometric Topology 25 (2025) 2613–2665

DOI: 10.2140/agt.2025.25.2613

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 $M T PL (d)$ and prove that there is a weak homotopy equivalence of the form $B {C o b}_{d}^{PL} ≃ Ω^{\infty - 1} M T PL (d)$ . We also discuss how to adjust the methods of this paper to obtain the topological version of our main result.

Keywords

cobordism categories, spaces of manifolds, Madsen–Tillmann spectra, PL topology

Mathematical Subject Classification

Primary: 57Q50, 57Q60

Secondary: 55P47, 55R40

References

Publication

Received: 10 August 2023

Revised: 10 June 2024

Accepted: 10 July 2024

Published: 17 September 2025

Authors

Mauricio Gómez López
Department of Mathematics and Computer Science University of Southern Denmark Odense Denmark

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Volume 25, issue 5 (2025)