Adiabatic groupoids and secondary invariants in K-theory

A geometric approach to K-homology for Lie manifolds

“…This is a compilation of known facts: the isomorphism A( ϕ G) ∼ = ϕ A is proved in [32], the Morita equivalence is proved in [22], that ϕ G gives a Lie structure on Σ is done in [39] and further developments can be found in [46].…”

“…Let (M, G) be a Lie manifold and ϕ : Σ → M a tame surjective submersion. The operations consisting of taking the adiabatic deformation and taking a pullback do not commute, however there are natural deformation groupoids relating them and it is exploited in [46] in order to obtain pushforward maps (which are there occurences of wrong way maps). We need to recall the results of [46] in our notation.…”

“…Using the restrictions homomorphisms ev u=i , i = 0, 1, and the natural Morita equivalence M we get homomorphisms (see [46] for more details):…”

“…Then the following result is essentially a rephrasing of [46]: Theorem 3.7. Let (M, G) be a Lie manifold and ϕ : Σ → M a tame surjective submersion.…”

Getzler rescaling via adiabatic deformation and a renormalized index formula

“…We introduce next a secondary invariant which encodes information in K-theory about the structure of a given compatible metric of positive scalar curvature. See also [52] where secondary invariants are introduced that control the vanishing of the generalized index defined via the adiabatic groupoid, instead of the Fredholm index defined via the Fredholm groupoid. Consider first the following general setup: Let A, B be separable C * -algebras.…”