Algebraic & Geometric Topology

A geometric interpretation of the homotopy groups of the cobordism category

Marcel Bökstedt and Anne Marie Svane

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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.

Article information

Algebr. Geom. Topol., Volume 14, Number 3 (2014), 1649-1676.

Received: 20 January 2013
Revised: 13 November 2013
Accepted: 20 November 2013
First available in Project Euclid: 19 December 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 57R90: Other types of cobordism [See also 55N22]
Secondary: 55Q05: Homotopy groups, general; sets of homotopy classes

cobordism categories classifying spaces Thom spectra fundamental group vector fields


Bökstedt, Marcel; Svane, Anne Marie. A geometric interpretation of the homotopy groups of the cobordism category. Algebr. Geom. Topol. 14 (2014), no. 3, 1649--1676. doi:10.2140/agt.2014.14.1649.

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