Positive scalar curvature and an equivariant Callias-type index theorem for proper actions

Equivariant K-theory, Groupoids and Proper Actions

In this paper we define complex equivariantK-theory for actions of Lie groupoids. For a Bredon-compatible Lie groupoid G, this defines a periodic cohomology theory on the category of finite G-CW-complexes. A suitable groupoid allows us to define complex equivariant K-theory for proper actions of non-compact Lie groups, which is a natural extension of the theory defined in [24]. For the particul...

A General Index Theorem for Callias-anghel Operators

We prove a families version of the index theorem for operators generalizing those studied by C. Callias and later by N. Anghel, which are operators on a manifold with boundary having the form D + iΦ, where D is elliptic pseudodifferential with self-adjoint symbols, and Φ is a self-adjoint bundle endomorphism which is invertible at the boundary and commutes with the symbol of D there. The index ...

Positive Scalar Curvature

One of the striking initial applications of the Seiberg-Witten invariants was to give new obstructions to the existence of Riemannian metrics of positive scalar curvature on 4– manifolds. The vanishing of the Seiberg–Witten invariants of a manifold admitting such a metric may be viewed as a non-linear generalization of the classic conditions [12, 11] derived from the Dirac operator. If a manifo...

An equivariant index formula for elliptic actions on contact manifolds

Given an elliptic action of a compact Lie group G on a co-oriented contact manifold (M, E) one obtains two naturally associated objects: A G-transversally elliptic operator Db / , and an equivariant differential form with generalised coefficients J (E, X) defined in terms of a choice of contact form on M . We explain how the form J (E, X) is natural with respect to the contact structure, and gi...

Positive Scalar Curvature and Surgery