On the Homotopy Classification of Elliptic Operators on Manifolds with Edges

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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    ABSTRACT: We find the stable homotopy classification of elliptic operators on stratified manifolds. Namely, we establish an isomorphism of the set of elliptic operators modulo stable homotopy and the $K$-homology group of the singular manifold. As a corollary, we obtain an explicit formula for the obstruction of Atiyah--Bott type to making interior elliptic operators Fredholm.
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    ABSTRACT: Dierential and pseudo-dierentia l operators on a manifold with (regular) geometric singularities can be studied within a calculus, in- spired by the concept of classical pseudo-dierentia l operators on a C1 manifold. In the singular case the operators form an algebra with a principal symbolic hierarchy = ( j)0 j k, with k being the order of the singularity and k operator-valued for k 1. The symbols de- termine ellipticity and the nature of parametrices. It is typical in this theory that, similarly as in boundary value problems (which are special edge problems, where the edge is just the boundary), there are trace, potential and Green operators, associated with the various strata of the configuration. The operators, obtained from the symbols by var- ious quantisations, act in weighted distribution spaces with multiple weights. We outline some essential elements of this calculus, give ex- amples and also comment on new challenges and interesting problems of the recent development.
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    ABSTRACT: The ellipticity of operators on a manifold with edge is defined as the bijectivity of the components of a principal symbolic hierarchy s = (sy,sÙ){\sigma=(\sigma_\psi,\sigma_\wedge)} , where the second component takes values in operators on the infinite model cone of the local wedges. In the general understanding of edge problems there are two basic aspects: Quantisation of edge-degenerate operators in weighted Sobolev spaces, and verifying the ellipticity of the principal edge symbol sÙ{\sigma_\wedge} which includes the (in general not explicitly known) number of additional conditions of trace and potential type on the edge. We focus here on these questions and give explicit answers for a wide class of elliptic operators that are connected with the ellipticity of edge boundary value problems and reductions to the boundary. In particular, we study the edge quantisation and ellipticity for Dirichlet–Neumann operators with respect to interfaces of some codimension on a boundary. We show analogues of the Agranovich–Dynin formula for edge boundary value problems.
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