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Index defects in the theory of nonlocal boundary value problems and the eta invariant
K-Theory and Homology · Submitted 15 Aug 2001
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim Thus, the two elliptic theories are dual.
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The abstract, sentence by sentence
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.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | We study elliptic theory on manifolds with boundary represented as a covering space. | contribution 0.70 | 0.71 | 0.02 | asserted 0.96 |
| s1 | Firstly, we consider boundary value problems, where the boundary conditions are allowed to mix the values of functions in the fibers of the covering. | contribution 0.94 | 0.50 | 0.03 | asserted 0.71 |
| s2 | We show that elliptic elements define Fredholm operators and prove an index formula. | result 0.86 | 0.96 | 0.17 | asserted 1.00 |
| s3 | For the identity covering, our formula reduces to the Atiyah-Bott index formula for classical boundary value problems. | result 0.62 | 0.66 | 0.06 | asserted 1.00 |
| s4 | Secondly, we consider Atiyah-Patodi-Singer boundary value problems for operators adapted to the covering. | contribution 0.94 | 0.66 | 0.03 | asserted 0.78 |
| s5 | We show that the corresponding symbols have a natural homotopy invariant. | result 0.90 | 0.91 | 0.05 | asserted 1.00 |
| s6 | This invariant is equal to the index of the corresponding spectral problem plus the relative eta-invariant. | result 1.00 | 0.83 | 0.15 | asserted 1.00 |
| s7 | The computation of this invariant in topological terms is one of the main results of the paper. | result 0.46 | 0.85 | 0.38 | asserted 1.00 |
| s8 | For a trivial covering, we recover the mod n-index theorem of Freed-Melrose. | result 0.82 | 0.77 | 0.08 | asserted 1.00 |
| s9 | Finally, we prove the Poincare duality and isomorphisms in K-theory of singular spaces corresponding to our manifolds. | result 0.64 | 0.93 | 0.15 | asserted 1.00 |
| s10 | The isomorphisms are defined in terms of the two classes of elliptic operators from the above. | contribution 0.37 | 0.88 | 0.13 | asserted 1.00 |
| s11 | Thus, the two elliptic theories are dual. | interpretation 0.92 | 0.82 | 0.32 | asserted 1.00 |
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Original 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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How the literature uses it
3 arXiv papers cite this in 4 sentences
Papers citing it · 1 sentence without a confident label
Uses its method or tool 1 paper
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To prove this, we consider the Calderon bundle [ref] (see also [ref] ) [equation] of the main operator in [ref].
Index of elliptic operators for a diffeomorphism · Homotopy of the boundary condition · 0.87
Cites it as support 1 paper
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The second was proved in [ref] .
Index formulas on stratified manifolds · Proof of theorem [ref] · 0.96
Cites it as background 1 paper
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Nonlocal boundary value problems of this type were considered in [ref] .
Index of elliptic operators for a diffeomorphism · Homotopy of the boundary condition · 0.89
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