A geometric interpretation of the homotopy groups of the cobordism category

TL;DR

This paper provides a geometric interpretation of the higher homotopy groups of the cobordism category's classifying space, relating them to cobordism groups with orthonormal tangent bundle sections.

Contribution

It introduces a geometric perspective on the higher homotopy groups of the cobordism category, extending previous classifications to include additional tangent bundle structure.

Findings

Higher homotopy groups correspond to cobordism groups with orthonormal tangent sections.

Fundamental group is a free group with geometrically intuitive relations.

Provides a geometric interpretation aligning with the classifying space's known structure.

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.