Tubular configurations: equivariant scanning and splitting

Abstract

Replacing configurations of points by configurations of tubular neighbourhoods (or discs) in a manifold $M$ 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 generalizations thereof that are equivariant. In particular, one deduces stable splittings of homotopy orbit spaces. As an application, the homology injectivity is proved for diffeomorphisms of $M$ that fix an increasing number of points. Throughout, we work with configurations spaces with labels in a fibre bundle over $M$.