The homotopy type of the PL cobordism category. I

2016/08/22 by Lopez, Mauricio Gomez
#Algebraic Topology (math.AT) #FOS: Mathematics #Geometric Topology (math.GT)

paper · doi:10.48550/arxiv.1608.06236

Abstract

In this paper, we introduce a bordism category 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 BC_d^PL is weak homotopy equivalent to an infinite loop space. We regard 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.

Related

• The homotopy type of the PL cobordism category. II
• Singular cobordism categories
• Cobordism Category of Manifolds With Baas-Sullivan Singularities, Part I
• The homotopy type of the topological cobordism category
• Cobordism categories of manifolds with Baas-Sullivan singularities, Part 2