Gysin maps and wrong way functoriality via geometric deformation groupoids

TL;DR

This paper develops a geometric framework using deformation groupoids to construct and unify pushforward maps in various (co)homology theories for Lie groupoids, including a novel approach to equivariant orbifold K-theory.

Contribution

It introduces deformation Lie groupoids to define pushforward maps functorially across (co)homology theories, generalizing previous results and including new cases like equivariant twisted orbifold K-theory.

Findings

Constructed deformation Lie groupoids for pushforward maps

Proved functoriality of these maps in (co)homology theories

Applied to equivariant twisted orbifold K-theory with groupoid actions

Abstract

In this article we study the normal bundle and the deformation to the normal cone functors to get deformation Lie groupoids that allow us to construct pushforward maps in any suitable (co)homology theory for Lie groupoids (not only K-theory) and in a natural and geometric way. The main theorems being the functoriality for these pushforward maps which recovers, unifies and generalizes many previous cases. The main new example we develop in this paper is the wrong way functoriality for equivariant (twisted) Orbifold K-theory with respect to a groupoid action.