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ISSN (electronic): 1472-2739
ISSN (print): 1472-2747
A geometric interpretation of the homotopy groups of the cobordism category

Marcel Bökstedt and Anne Marie Svane

Algebraic & Geometric Topology 14 (2014) 1649–1676
Abstract

The classifying space of the embedded cobordism category has been identified by Galatius, Tillmann, Madsen and Weiss [Acta. Math. 202 (2009) 195–239] as the infinite loop space of a certain Thom spectrum. This identifies the set of path components with the classical cobordism group. In this paper, we give a geometric interpretation of the higher homotopy groups as certain cobordism groups where all manifolds are now equipped with a set of orthonormal sections in the tangent bundle. We also give a description of the fundamental group as a free group with a set of geometrically intuitive relations.

Keywords
cobordism categories, classifying spaces, Thom spectra, fundamental group, vector fields
Mathematical Subject Classification 2010
Primary: 57R90
Secondary: 55Q05
References
Publication
Received: 20 January 2013
Revised: 13 November 2013
Accepted: 20 November 2013
Published: 29 May 2014
Authors
Marcel Bökstedt
Department of Mathematics
Aarhus University
Ny Munkegade 118
DK-8000 Aarhus C
Denmark
Anne Marie Svane
Department of Mathematics
Aarhus University
Ny Munkegade 118
DK-8000 Aarhus C
Denmark