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

Source: arXiv


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]. "
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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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