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The index problem on manifolds with edges. (English) Zbl 1115.58022
Fuchs, Jürgen (ed.) et al., Noncommutative geometry and representation theory in mathematical physics. Satellite conference to the fourth European congress of mathematics, July 5–10, 2004, Karlstad, Sweden. Providence, RI: American Mathematical Society (AMS) (ISBN 0-8218-3718-4/pbk). Contemporary Mathematics 391, 289-312 (2005).
The authors introduce differential operators on manifolds with edges, algebraic and geometric symbols, Sobolev completions of function spaces, and finiteness theorem for elliptic operators. A manifold \(\mathcal M\) with edges is a stratified set consisting of the interior stratum \(M^\circ\) and the singular stratum \(X\). They investigate the elliptic operators on \(\mathcal M\) which have the indices as the index (contribution of the principal symbol) on \(M^\circ\) plus the index (contribution of the edge symbol) on \(X\), called index splitting.
They show that there is an obstruction for an elliptic operator to index splitting, and single out the condition for the index splitting. There are a number of such classes that can be described by symmetry conditions. They establish an index theorem for such elliptic edge-degenerate operators.
For the entire collection see [Zbl 1078.17001].
MSC:
58J20 Index theory and related fixed-point theorems on manifolds
19K56 Index theory
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