Tubular configurations: equivariant scanning and splitting

TL;DR

This paper introduces equivariant scanning and splitting maps for configuration spaces of tubular neighborhoods in manifolds, leading to stable splittings and homology injectivity results under diffeomorphism actions.

Contribution

It develops new equivariant maps and stable splitting techniques for configuration spaces with labels, extending classical results to a diffeomorphism-equivariant setting.

Findings

Constructed a natural equivariant scanning map for tubular configurations.

Established stable splittings of homotopy orbit spaces.

Proved homology injectivity for diffeomorphisms fixing increasing points.

Abstract

Replacing configurations of points by configurations of tubular neighbourhoods (or discs) in a manifold, we are able to define a natural scanning map that is equivariant under the action of the diffeomorphism group of the manifold. We also construct the so-called power set map of configuration spaces diffeomorphism equivariantly. Combining these two constructions yields stable splittings in the sense of Snaith and generalisations thereof that are equivariant. In particular one deduces stable splittings of homotopy orbit spaces. As an application the homology injectivity is proved for diffeomorphism of manifolds that fix an increasing number of points. Throughout we work with configurations spaces with labels in a fibre bundle.