1. Higher genera for proper actions of Lie groups
- Author
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Hessel Posthuma, Paolo Piazza, and Algebra, Geometry & Mathematical Physics (KDV, FNWI)
- Subjects
Mathematics - Differential Geometry ,Pure mathematics ,positive scalar curvature ,$K\mkern-2mu$-theory ,higher signatures ,Cyclic homology ,Assessment and Diagnosis ,01 natural sciences ,higher genera ,group cocycles ,index classes ,0103 physical sciences ,FOS: Mathematics ,proper actions ,Sectional curvature ,0101 mathematics ,cyclic cohomology ,Quotient ,higher index formulae ,Mathematics ,19K56 ,Lie groups ,higher indices ,010102 general mathematics ,Lie group ,K-Theory and Homology (math.KT) ,G-homotopy invariance ,$G$-homotopy invariance ,Manifold ,58J20 ,58J42 ,van Est isomorphism ,Differential Geometry (math.DG) ,Metric (mathematics) ,Mathematics - K-Theory and Homology ,010307 mathematical physics ,Geometry and Topology ,Analysis ,Maximal compact subgroup ,Scalar curvature - 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 non-positive 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. In this paper 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-hat genera of a G-spin, G-proper manifold admitting a G-invariant metric of positive scalar curvature., 20 pages, revised version, the main changes are in section 2.3
- Published
- 2019