Geodesic
A geometric interpretation of the homotopy groups of the cobordism category
Bökstedt, Marcel1 ; Svane, Anne Marie1
1Department of Mathematics, Aarhus University, Ny Munkegade 118, DK-8000 Aarhus C, Denmark
Algebraic and Geometric Topology, Tome 14 (2014) no. 3, pp. 1649-1676
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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.

DOI : 10.2140/agt.2014.14.1649
Classification : 57R90, 55Q05
Keywords: cobordism categories, classifying spaces, Thom spectra, fundamental group, vector fields

Bökstedt, Marcel  1   ; Svane, Anne Marie  1

1 Department of Mathematics, Aarhus University, Ny Munkegade 118, DK-8000 Aarhus C, Denmark 
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Bökstedt, Marcel; Svane, Anne Marie. A geometric interpretation of the homotopy groups of the cobordism category. Algebraic and Geometric Topology, Tome 14 (2014) no. 3, pp. 1649-1676. doi: 10.2140/agt.2014.14.1649

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