Positive scalar curvature and Poincaré duality for proper actions

Abstract

For G an almost-connected Lie group, we study G-equivariant index theory for proper co-compact actions with various applications, including obstructions to and existence of G-invariant Riemannian metrics of positive scalar curvature. We prove a rigidity result for almost-complex manifolds, generalising Hattori's results, and an analogue of Petrie's conjecture. When G is an almost-connected Lie group or a discrete group, we establish Poincaré duality between G-equivariant K-homology and K-theory, observing that Poincaré duality does not necessarily hold for general G.

Index of Equivariant Callias-Type Operators and Invariant Metrics of Positive Scalar Curvature

“…The methods and results of this paper can be contrasted with the equivariant index theory and applications studied in [14]. First, [14] deals exclusively with index theory in the cocompact case, where the C * (G)-Fredholmness of the Dirac operator was known. In addition, [14] focused almost entirely on the case of almost-connected G, where a global slice of the manifold exists.…”

Positive Scalar Curvature and Callias-Type Index Theorems for Proper Actions

“…In the setting of Theorem 2.9, Theorem 6.2 implies that N admits a K-invariant metric with uniformly positive scalar curvature. By Theorem 4.6 in [17] (see also Theorem 58 in [18]), this metric induces a G-invariant Riemannian metric on G × K N of uniformly positive scalar curvature.…”

“…Suppose that G is an almost connected, reductive Lie group, and let K < G be maximal compact. In [18,19,24] some results were proved relating G-equivariant indices to K-equivariant indices via Dirac induction. Such results allow one to deduce results in equivariant index theory for actions by noncompact groups from corresponding results for compact groups.…”

“…Such results allow one to deduce results in equivariant index theory for actions by noncompact groups from corresponding results for compact groups. This was applied to obtain results in geometric quantisation [19,20,22] and geometry of group actions [18,21]. Corollary 7.4 below is a version of this idea for the index of Definition 3.3.…”

“…We consider the case where G is not necessarily discrete, and does not necessarily act freely. Results on this case of Question 1.1 in the case where G is an almost-connected Lie group and M/G is compact were obtained in [18,34]. In this paper, we focus on the case where M/G is noncompact.…”