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- K-theory (7)
- elliptic operators (6)
- relative index (6)
- Atiyah-Patodi-Singer theory (5)
- Fredholm property (5)
- index theory (5)
- boundary value problems (4)
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- index (4)
- manifold with singularities (4)
- surgery (4)
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- conormal symbol (3)
- eta invariant (3)
- index of elliptic operators in subspaces (3)
- Atiyah-Bott condition (2)
- Atiyah-Bott obstruction (2)
- Fredholm operators (2)
- Lefschetz fixed point formula (2)
- Mellin transform (2)
- boundary value problem (2)
- edge-degenerate operators (2)
- elliptic boundary value problems (2)
- elliptic families (2)
- elliptic family (2)
- ellipticity (2)
- eta-invariant (2)
- homotopy classification (2)
- index formulas (2)
- linking coefficients (2)
- manifolds with conical singularities (2)
- modn-index (2)
- pseudodiferential operators (2)
- quantization (2)
- regularizer (2)
- regularizers (2)
- spectral flow (2)
- symmetry conditions (2)
- (co)boundary operator (1)
- APS problem (1)
- Atiyah-Singer theorem (1)
- Calderón projections (1)
- Cauchy Riemann operator (1)
- Chern character (1)
- Dirac operators (1)
- Euler operator (1)
- Green operator (1)
- Pontrjagin duality (1)
- Riemann-Roch theorem (1)
- Sobolev problem (1)
- analytic index (1)
- contact transformations (1)
- covering (1)
- dimension functional (1)
- edge symbol (1)
- elliptic morphism (1)
- elliptic operators in subspaces (1)
- elliptic problem (1)
- exterior tensor product (1)
- finiteness theorem (1)
- index formula (1)
- index of elliptic operator (1)
- manifold with edge (1)
- manifolds with edges (1)
- mod k index (1)
- modulo n index (1)
- nonhomogeneous boundary value problems (1)
- nonlocal problem (1)
- parameter-dependent ellipticity (1)
- parity condition (1)
- parity conditions (1)
- problem of classification (1)
- pseudo-diferential operators (1)
- pseudodifferential subspace (1)
- pseudodifferential subspaces (1)
- relative η-invariant (1)
- spectral boundary value problems (1)
- spectral resolution (1)
- symplectic (canonical) transformations (1)
- η-invariant (1)
Institute
The topological significance of the spectral Atiyah-Patodi-Singer η-invariant is investigated. We show that twice the fractional part of the invariant is computed by the linking pairing in K-theory with the orientation bundle of the manifold. The Pontrjagin duality implies the nondegeneracy of the linking form. An example of a nontrivial fractional part for an even-order operator is presented.
We prove a general theorem on the behavior of the relative index under surgery for a wide class of Fredholm operators, including relative index theorems for elliptic operators due to Gromov-Lawson, Anghel, Teleman, Booß-Bavnbek-Wojciechowski, et al. as special cases. In conjunction with additional conditions (like symmetry conditions), this theorem permits one to compute the analytical index of a given operator. In particular, we obtain new index formulas for elliptic pseudodifferential operators and quantized canonical transformations on manifolds with conical singularities.
Relative elliptic theory
(2002)
This paper is a survey of relative elliptic theory (i.e. elliptic theory in the category of smooth embeddings), closely related to the Sobolev problem, first studied by Sternin in the 1960s. We consider both analytic aspects to the theory (the structure of the algebra of morphismus, ellipticity, Fredholm property) and topological aspects (index formulas and Riemann-Roch theorems). We also study the algebra of Green operators arising as a subalgebra of the algebra of morphisms.
We prove a general theorem on the local property of the relative index for a wide class of Fredholm operators, including relative index theorems for elliptic operators due to Gromov-Lawson, Anghel, Teleman, Booß-Bavnbek-Wojciechowski, et al. as special cases. In conjunction with additional conditions (like symmetry conditions) this theorem permits one to compute the analytical index of a given operator. In particular, we obtain new index formulas for elliptic pseudodifferential operators and quantized canonical transformations on manifolds with conical singularities as well as for elliptic boundary value problems with a symmetry condition for the conormal symbol.
An elliptic theory is constructed for operators acting in subspaces defined via even pseudodifferential projections. Index formulas are obtained for operators on compact manifolds without boundary and for general boundary value problems. A connection with Gilkey's theory of η-invariants is established.
The aim of this paper is to explain the notion of subspace defined by means of pseudodifferential projection and give its applications in elliptic theory. Such subspaces are indispensable in the theory of well-posed boundary value problems for an arbitrary elliptic operator, including the Dirac operator, which has no classical boundary value problems. Pseudodifferential subspaces can be used to compute the fractional part of the spectral Atiyah–Patodi–Singer eta invariant, when it defines a homotopy invariant (Gilkey’s problem). Finally, we explain how pseudodifferential subspaces can be used to give an analytic realization of the topological K-group with finite coefficients in terms of elliptic operators. It turns out that all three applications are based on a theory of elliptic operators on closed manifolds acting in subspaces.
An elliptic theory is constructed for operators acting in subspaces defined via even pseudodifferential projections. Index formulas are obtained for operators on compact manifolds without boundary and for general boundary value problems. A connection with Gilkey's theory of η-invariants is established.