An index formula on manifolds with fibered cusp ends

Details An index formula on manifolds with fibered cusp ends An index formula on manifolds with fibered cusp ends

2002-12-17

arxiv.org/abs/math/0212239v1

J. Geom. Analysis 15 (2005), 261-283

Mathematics

Differential Geometry

22 pages. Prepublication du Laboratoire Emile Picard n.253. See also http://picard.ups-tlse.fr

Scientific paper

We consider a compact manifold whose boundary is a locally trivial fiber bundle and an associated pseudodifferential algebra that models fibered cusps at infinity. Using trace-like functionals that generate the 0-dimensional Hochschild cohomology groups, we express the index of a fully elliptic fibered cusp operator as the sum of a local contribution from the interior and a term that comes from the boundary. This answers the index problem formulated by Mazzeo and Melrose. We give a more precise answer in the case where the base of the boundary fiber bundle is the circle. In particular, for Dirac operators associated to a "product fibered cusp metric", the index is given by the integral of the Atiyah-Singer form in the interior minus the adiabatic limit of the eta invariant of the restriction of the operator to the boundary.

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