The geometry of the local cohomology filtration in equivariant bordism

Abstract

We present geometric constructions which realize the local cohomology filtration in the setting of equivariant bordism, with the aim of understanding free $G$ actions on manifolds. We begin by reviewing the basic construction of the local cohomology filtration, starting with the Conner-Floyd tom Dieck exact sequence. We define this filtration geometrically using the language of families of subgroups. We then review Atiyah-Segal-Wilson $K$-theory invariants, which are well-suited for studying the manifolds produced by our techniques. We end by indicating potential applications of these ideas.