Vol. 6, No. 4, 2021

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Higher genera for proper actions of Lie groups, II: The case of manifolds with boundary

Paolo Piazza and Hessel B. Posthuma

Vol. 6 (2021), No. 4, 713–782
DOI: 10.2140/akt.2021.6.713
Abstract

Let G be a finitely connected Lie group and let K be a maximal compact subgroup. Let M be a cocompact G-proper manifold with boundary, endowed with a G-invariant metric which is of product type near the boundary. Under additional assumptions on G, for example that it satisfies the rapid decay condition and is such that GK has nonpositive sectional curvature, we define higher Atiyah–Patodi–Singer C-indices associated to elements [φ] Hdiff(G) and to a generalized G-equivariant Dirac operator D on M with L2-invertible boundary operator D. We then establish a higher index formula for these C-indices and use it in order to introduce higher genera for M, thus generalizing to manifolds with boundary the results that we have established in Part I. Our results apply in particular to a semisimple Lie group G. We use crucially the pairing between suitable relative cyclic cohomology groups and relative K-theory groups.

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Keywords
Atiyah–Patodi–Singer higher index theory, higher indices, $K\mkern-2mu$-theory, cyclic cohomology, Lie groups, proper actions, noncommutative geometry, groupoids, group cocycles, delocalized cocycles, index classes, relative pairing, excision
Mathematical Subject Classification
Primary: 58J20
Secondary: 19K56, 58J22, 58J42
Milestones
Received: 1 February 2021
Revised: 1 June 2021
Accepted: 21 June 2021
Published: 12 February 2022
Authors
Paolo Piazza
Dipartimento di Matematica
Università degli Studi di Roma “La Sapienza”
Rome
Italy
Hessel B. Posthuma
University of Amsterdam
Amsterdam
Netherlands