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2022 Higher genera for proper actions of Lie groups II: The case of manifolds with boundary
Paolo Piazza, Hessel B. Posthuma
Ann. K-Theory 6(4): 713-782 (2022). DOI: 10.2140/akt.2021.6.713


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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Paolo Piazza. Hessel B. Posthuma. "Higher genera for proper actions of Lie groups II: The case of manifolds with boundary." Ann. K-Theory 6 (4) 713 - 782, 2022.


Received: 1 February 2021; Revised: 1 June 2021; Accepted: 21 June 2021; Published: 2022
First available in Project Euclid: 8 May 2022

Digital Object Identifier: 10.2140/akt.2021.6.713

Primary: 58J20
Secondary: 19K56 , 58J22 , 58J42

Keywords: Atiyah–Patodi–Singer higher index theory , cyclic cohomology , delocalized cocycles , excision , group cocycles , groupoids , higher indices , index classes , ‎K-theory , Lie groups , noncommutative geometry , proper actions , relative pairing

Rights: Copyright © 2022 Mathematical Sciences Publishers


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Vol.6 • No. 4 • 2022
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