Equivariant embedding theorems and topological index maps

TL;DR

This paper develops a framework for constructing topological index maps for equivariant Dirac operators using normally non-singular maps, generalizing embedding theorems and connecting to KK-theory.

Contribution

It introduces the concept of normally non-singular maps for equivariant settings, generalizes the Mostow Embedding Theorem, and links these maps to topological index theory and KK-theory.

Findings

Normal factorisations exist for smooth maps under certain conditions.

Equivariant normally non-singular maps induce wrong-way maps in cohomology.

The functor from K-oriented maps to KK-theory models the topological index map.

Abstract

The construction of topological index maps for equivariant families of Dirac operators requires factoring a general smooth map through maps of a very simple type: zero sections of vector bundles, open embeddings, and vector bundle projections. Roughly speaking, a normally non-singular map is a map together with such a factorisation. These factorisations are models for the topological index map. Under some assumptions concerning the existence of equivariant vector bundles, any smooth map admits a normal factorisation, and two such factorisations are unique up to a certain notion of equivalence. To prove this, we generalise the Mostow Embedding Theorem to spaces equipped with proper groupoid actions. We also discuss orientations of normally non-singular maps with respect to a cohomology theory and show that oriented normally non-singular maps induce wrong-way maps on the chosen cohomology theory. For K-oriented normally non-singular maps, we also get a functor to Kasparov's equivariant KK-theory. We interpret this functor as a topological index map.