Configuration categories and homotopy automorphisms

TL;DR

This paper explores how the configuration category of a manifold's interior encodes the homotopical structure of the manifold with boundary, revealing an action of homotopy automorphisms that aligns with boundary homeomorphisms.

Contribution

It introduces a homotopical model for manifolds with boundary derived from configuration categories, linking automorphisms to boundary homeomorphisms.

Findings

Homotopical model recovers (M, boundary of M) from configuration category.

Derived homotopy automorphisms act on the model compatibly with boundary homeomorphisms.

Establishes a connection between automorphisms of configuration categories and manifold symmetries.

Abstract

Let M be a smooth compact manifold with boundary. Under some geometric conditions on M, a homotopical model for the pair (M,boundary of M) can be recovered from the configuration category of the interior of M. The grouplike monoid of derived homotopy automorphisms of the configuration category of the interior of M then acts on the homotopical model of the pair (M,boundary of M), in a way which is compatible with a known homotopical action of the homeomorphism group of M minus boundary on (M,boundary of M).