Article

# Index defects in the theory of nonlocal boundary value problems and the eta invariant

09/2001;
Source: arXiv

ABSTRACT We study elliptic theory on manifolds with boundary represented as a covering space. Firstly, we consider boundary value problems, where the boundary conditions are allowed to mix the values of functions in the fibers of the covering. We show that elliptic elements define Fredholm operators and prove an index formula. For the identity covering, our formula reduces to the Atiyah-Bott index formula for classical boundary value problems. Secondly, we consider Atiyah-Patodi-Singer boundary value problems for operators adapted to the covering. We show that the corresponding symbols have a natural homotopy invariant. This invariant is equal to the index of the corresponding spectral problem plus the relative eta-invariant. The computation of this invariant in topological terms is one of the main results of the paper. For a trivial covering, we recover the mod n-index theorem of Freed-Melrose. Finally, we prove the Poincare duality and isomorphisms in K-theory of singular spaces corresponding to our manifolds. The isomorphisms are defined in terms of the two classes of elliptic operators from the above. Thus, the two elliptic theories are dual.

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Available from: Anton Savin, Apr 22, 2013
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• "Boundary value problems similar to Carleman's problem, with the boundary condition relating the values of the unknown function at different points on the boundary were considered (see monograph of Antonevich, Belousov and Lebedev [1] and the references cited there). Finiteness theorems were proved and for the case of finite group actions index theorems were obtained (see also [48]). On the other hand, nonlocal boundary value problems, in which the boundary condition relates the values of a function on the boundary of the domain and on submanifolds, which lie inside the domain there were considered in [15, 62–64]. "
##### Article: Elliptic theory for operators associated with diffeomorphisms of smooth manifolds
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ABSTRACT: In this paper we give a survey of elliptic theory for operators associated with diffeomorphisms of smooth manifolds. Such operators appear naturally in analysis, geometry and mathematical physics. We survey classical results as well as results obtained recently. The paper consists of an introduction and three sections. In the introduction we give a general overview of the area of research. For the reader's convenience here we tried to keep special terminology to a minimum. In the remaining sections we give detailed formulations of the most important results mentioned in the introduction.
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• "It should be mentioned that the importance and the interest of this class of boundary value problems on manifolds with fibered boundary is clear even in the case, when the boundary is a covering, i.e. a fibration with a discrete fiber. For a covering, the class of boundary value problems under consideration includes a number of nonlocal boundary value problems (see [2]). Rather surprisingly, the boundary conditions in the intermediate theories on manifolds with fibered boundary are defined by operators with discontinuous symbols. "
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ABSTRACT: We define a class of boundary value problems on manifolds with fibered boundary. This class is in a certain sense a deformation between the classical boundary value problems and the Atiyah–Patodi–Singer problems in subspaces (it contains both as special cases). The boundary conditions in this theory are taken as elements of the C *-algebra generated by pseudodifferential operators and families of pseudodifferential operators in the fibers. We prove the Fredholm property for elliptic boundary value problems and compute a topological obstruction (similar to Atiyah–Bott obstruction) to the existence of elliptic boundary conditions for a given elliptic operator. Geometric operators with trivial and nontrivial obstruction are given. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)
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