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) over cap -genera of a G-spin G-proper manifold admitting a G-invariant metric of positive scalar curvature.
Higher genera for proper actions of Lie groups / Piazza, Paolo; Posthuma, Hessel B.. - In: THE ANNALS OF K-THEORY. - ISSN 2379-1683. - 4:3(2019), pp. 473-504. [10.2140/akt.2019.4.473]
Higher genera for proper actions of Lie groups
Piazza, Paolo
;
2019
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) over cap -genera of a G-spin G-proper manifold admitting a G-invariant metric of positive scalar curvature.File | Dimensione | Formato | |
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