Vol. 4, No. 3, 2019

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Higher genera for proper actions of Lie groups

Paolo Piazza and Hessel B. Posthuma

Vol. 4 (2019), No. 3, 473–504
Abstract

Let G be a Lie group with finitely many connected components and let K be a maximal compact subgroup. We assume that G satisfies the rapid decay (RD) property and that GK has a nonpositive sectional curvature. As an example, we can take G to be a connected semisimple Lie group. Let M be a G-proper manifold with compact quotient MG. Building on work by Connes and Moscovici (1990) and Pflaum et al. (2015), we establish index formulae for the C-higher indices of a G-equivariant Dirac-type operator on M. We use these formulae to investigate geometric properties of suitably defined higher genera on M. In particular, we establish the G-homotopy invariance of the higher signatures of a G-proper manifold and the vanishing of the Â-genera of a G-spin G-proper manifold admitting a G-invariant metric of positive scalar curvature.

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Keywords
Lie groups, proper actions, group cocycles, van Est isomorphism, cyclic cohomology, $K\mkern-2mu$-theory, index classes, higher indices, higher index formulae, higher signatures, $G$-homotopy invariance, higher genera, positive scalar curvature
Mathematical Subject Classification 2010
Primary: 58J20
Secondary: 19K56, 58J42
Milestones
Received: 19 June 2018
Revised: 21 February 2019
Accepted: 12 March 2019
Published: 17 December 2019
Authors
Paolo Piazza
Dipartimento di Matematica
Università degli Studi di Roma “La Sapienza”
Roma
Italy
Hessel B. Posthuma
Kortweg–de Vries Institute for Mathematics
University of Amsterdam
Amsterdam
Netherlands