Equivariant embedding theorems and topological index maps

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.

References in corpus (4)

• On the Equivalence of Geometric and Analytic K-Homology
• Equivariant representable K-theory
• Bivariant K-theory via correspondences
• Dualities in equivariant Kasparov theory

Cited by in corpus (6)

• Bivariant K-theory via correspondences
• Exhausting families of representations and spectra of pseudodifferential operators
• Quotient Theorems in Polyfold Theory and $S^1$-Equivariant Transversality
• Geometric construction of classes in van Daele's K-theory
• Baum-Connes and the Fourier-Mukai transform
• Motivic Bivariant Characteristic Classes