A geometric interpretation of the homotopy groups of the cobordism category

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.

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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

Volume 14, issue 3 (2014)

Marcel Bökstedt and Anne Marie Svane

Abstract