Equivariant embedding theorems and topological index maps

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.

💡 Research Summary

The paper develops a unified geometric framework for constructing topological index maps for equivariant families of Dirac operators. The central notion is that of a “normally non‑singular map”: a smooth G‑equivariant map f : X → Y together with a factorisation f = p ∘ u ∘ s, where s : X → E is the zero‑section of a G‑vector bundle E → X, u : E → F is a G‑equivariant open embedding into another G‑vector bundle F → Y, and p : F → Y is the bundle projection. This three‑step factorisation replaces the classical embedding‑Thom‑push‑forward recipe and works uniformly for actions of proper Lie groupoids.

The authors first prove an existence theorem: under the hypothesis that the underlying proper groupoid admits enough equivariant vector bundles (a condition satisfied for most proper Lie groupoids), any smooth equivariant map admits a normal factorisation. The proof is a proper‑groupoid version of the Mostow embedding theorem. One embeds X equivariantly into a large enough G‑vector bundle V → X, then uses a G‑invariant metric and connection to obtain a G‑equivariant immersion into a bundle over Y, which after a small perturbation becomes an open embedding u. The zero‑section and projection complete the factorisation.

A uniqueness result follows: any two normal factorisations of the same map are equivalent up to G‑equivariant bundle isomorphisms and homotopies of the open embedding. This establishes a well‑defined “normal class” of the map.

Next, the paper introduces orientations with respect to a generalized cohomology theory E. An E‑orientation of a normally non‑singular map is a Thom isomorphism for the involved bundles that is compatible with the open embedding. When such an orientation exists, the factorisation yields a canonical wrong‑way map  f! : E⁎(Y) → E⁎⁺ⁿ(X) (where n is the virtual bundle dimension). The construction respects composition and is natural with respect to equivariant maps.

Specialising to K‑theory, an E‑orientation becomes a K‑orientation, and the wrong‑way map coincides with the Kasparov product class