Index of equivariant Callias-type operators and invariant metrics of positive scalar curvature (Q2658988)

For any Lie group \(G\) acting isometrically on a manifold \(M\), the author formulates the general notion of a \(G\)-equivariant elliptic operator that is invertible outside of a \(G\)-cocompact subset of \(M\). To establish the analogue of the Rellich lemma in this setting, the author defines \(G\)-Sobolev modules from the \(G\)-action on the space of compactly supported smooth section. It is shown that \(G\)-Callias-type operators are self-adjoint, regular in the sense of Hilbert modules and hence equivariantly invertible at infinity. The paper also gives an explicit construction of \(G\)-Callias-type operator using \(K\)-theory of an equivariant Higson corona of \(M\). Finally the paper obtains an obstruction to positive scalar curvature metrics on non-cocompact manifolds as an application of the theory.