The homotopy type of the PL cobordism category, I

Gómez López, Mauricio

doi:10.2140/agt.2025.25.2527

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The homotopy type of the PL cobordism category, I

Mauricio Gómez López

Algebraic & Geometric Topology 25 (2025) 2527–2611

DOI: 10.2140/agt.2025.25.2527

Abstract

We introduce a bordism category ${C o b}_{d}^{PL}$ 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, we show that the classifying space $B {C o b}_{d}^{PL}$ is weak homotopy equivalent to an infinite loop space. We regard ${C o b}_{d}^{PL}$ as the piecewise linear analogue of the category of smooth cobordisms which has been studied extensively in connection with the Madsen–Weiss theorem, and our main result is a first step towards obtaining Madsen–Weiss type results in the context of PL topology.

Keywords

cobordism categories, spaces of manifolds, PL topology

Mathematical Subject Classification 2010

Primary: 57Q20

Secondary: 55P47

References

Publication

Received: 28 October 2016

Revised: 11 July 2024

Accepted: 18 September 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)