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2019 Higher genera for proper actions of Lie groups
Paolo Piazza, Hessel B. Posthuma
Ann. K-Theory 4(3): 473-504 (2019). DOI: 10.2140/akt.2019.4.473

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 G K 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 M G . 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 A ̂ -genera of a G -spin G -proper manifold admitting a G -invariant metric of positive scalar curvature.

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Paolo Piazza. Hessel B. Posthuma. "Higher genera for proper actions of Lie groups." Ann. K-Theory 4 (3) 473 - 504, 2019. https://doi.org/10.2140/akt.2019.4.473

Information

Received: 19 June 2018; Revised: 21 February 2019; Accepted: 12 March 2019; Published: 2019
First available in Project Euclid: 3 January 2020

zbMATH: 07146017
MathSciNet: MR4043466
Digital Object Identifier: 10.2140/akt.2019.4.473

Subjects:
Primary: 58J20
Secondary: 19K56 , 58J42

Keywords: $G$-homotopy invariance , $K\mkern-2mu$-theory , cyclic cohomology , group cocycles , higher genera , higher index formulae , higher indices , higher signatures , index classes , Lie groups , positive scalar curvature , proper actions , van Est isomorphism

Rights: Copyright © 2019 Mathematical Sciences Publishers

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Vol.4 • No. 3 • 2019
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