Eric Leichtnam, Paolo Piazza
Étale groupoids, eta invariants and index theory
Let ? be a discrete finitely generated group. Let ![]()
? T be a ?-equivariant fibration, with fibers diffeomorphic to a fixed
even dimensional manifold with boundary Z. We assume that ? ? ![]()
? ![]()
| ? is a Galois covering of a compact manifold
with boundary. Let (D +(?))? ? T be a ?-equivariant family of Dirac-type operators. Under the assumption that the
boundary family is L2-invertible, we define an index class
in K0(C 0(T ) ?r ? ).
If, in addition, ? is of polynomial growth, we define higher indices by pairing
the index class with suitable cyclic cocycles. Our main result is then a formula
for these higher indices: the structure of the formula is as in the seminal work
of Atiyah, Patodi and Singer, with an interior geometric contribution and a
boundary contribution in the form of a higher eta invariant associated to the
boundary family. Under similar assumptions we extend our theorem to any
G -proper manifold, with G an étale groupoid. We employ this
generalization in order to establish a higher Atiyah-Patodi-Singer index formula
on certain foliations with boundary. Fundamental to our work is a suitable
generalization of Melrose b -pseudodifferential calculus as well as the
superconnection proof of the index theorem on G -proper manifolds recently given by Gorokhovsky and Lott in [A. Gorokhovsky and J. Lott , Local index theory over étale groupoids, J. reine angew. Math. 560
(2003), 151–198].
Journal fur die reine und angewandte Mathematik (Crelles Journal), Walter de Gruyter
Print ISSN: 0075-4102
Volume: 2005, 10/2005
Seiten: 169 - 233
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