Article

# On the Homotopy Classification of Elliptic Operators on Manifolds with Edges

04/2005;
Source: arXiv

ABSTRACT

We obtain a classification of elliptic operators modulo stable homotopy on manifolds with edges (this is in some sense the simplest class of manifolds with nonisolated singularities). We show that the operators are classified by the K-homology group of the manifold. To this end, we construct exact sequence in elliptic theory isomorphic to the K-homology exact sequence. The main part of the proof is the computation of the boundary map using semiclassical quantization.

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Available from: Vladimir Nazaikinskii, Apr 15, 2013
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• "Ell(X) ≃ K 0 (X) (1) on any compact stratified manifold X with arbitrary number of strata, where Ell(X) is the group generated by elliptic pseudodifferential operators on X modulo stable homotopy. Special cases of such isomorphisms were obtained in [5] [6] [7] [8] [9] for manifolds with two strata. The isomorphism (1) enables one to apply topological methods of K-homology theory in elliptic theory. "
##### Article: Homotopy classification of elliptic operators on stratified manifolds
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ABSTRACT: The problem of homotopy classification of elliptic operators on an arbitrary stratified manifold is discussed. A classification of elliptic operators on smooth manifolds up to homotopy is an essential point in the solution of the index problem by Atiyah and Singer and the classification is in terms of the K homology of a stratified manifold. The ideas of Atiyah allow to avoid the difficulties caused by the fact that the In the situation under consideration,the operators module compact operators are determined by whole sets of symbols of strata rather than by only one symbol, and the ellipticity condition is of an infinite-dimensional character. As application an index formula and a topological obstruction for Fredholm problems is obtained and to calculate the K group algebras of pseudodifferential operators(PDOs). Stable homotopy is an equivalence on the set of elliptic PDOs acting on sections of bundles.
Doklady Mathematics 08/2006; 73(3). DOI:10.1070/IM2007v071n06ABEH002386 · 0.38 Impact Factor
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• "For the case of isolated singularities, the fact that (0.1) is an isomorphism was proved in [2]. (See also [4], where the case of one singular point was considered.) The classification (0.1) has many corollaries and applications, including a formula for the obstruction, similar to the Atiyah–Bott obstruction [5], to the existence of Fredholm problems for elliptic equations, the fact that the group Ell 0 (M) is equal modulo torsion to the homology H ev (M), a generalization of Poincaré duality in K-theory to manifolds with singularities, etc. "
##### Article: On the Homotopy Classification of Elliptic Operators on Manifolds with Edges
[Hide abstract]
ABSTRACT: We obtain a classification of elliptic operators modulo stable homotopy on manifolds with edges (this is in some sense the simplest class of manifolds with nonisolated singularities). We show that the operators are classified by the K-homology group of the manifold. To this end, we construct exact sequence in elliptic theory isomorphic to the K-homology exact sequence. The main part of the proof is the computation of the boundary map using semiclassical quantization.
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• "Added in proof. The homotopy classification of edge-degenerate elliptic operators on manifolds with edges was obtained in [16] using the methods of the present paper. "
##### Article: Elliptic Operators on Manifolds with Singularities and K-Homology
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ABSTRACT: Elliptic operators on smooth compact manifolds are classified by K-homology. We prove that a similar classification is valid also for manifolds with simplest singularities: isolated conical points and edges. The main ingredients of the proof of these results are: Atiyah–Singer difference construction in the noncommutative case and Poincar isomorphism in K-theory for (our) singular manifolds. As an application we give a formula in topological terms for the obstruction to Fredholm problems on manifolds with edges.
K-Theory 12/2004; 34(1):71-98. DOI:10.1007/s10977-005-1515-1 · 1.13 Impact Factor