A geometric interpretation of the homotopy groups of the cobordism category

Abstract

The classifying space of the embedded cobordism category has been identified in by Galatius, Tillmann, Madsen, and Weiss 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.

23 pages

References in corpus (1)

• Monoids of moduli spaces of manifolds

Cited by in corpus (3)

• On the simplicial volume and the Euler characteristic of (aspherical) manifolds
• Bosonization and Anomaly Indicators of (2+1)-D Fermionic Topological Orders
• Semisimple Field Theories Detect Stable Diffeomorphism