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Homotopy classification of elliptic operators on stratified manifolds. (English. Russian original) Zbl 1327.58025
Dokl. Math. 73, No. 3, 407-411 (2006); translation from Dokl. Akad. Nauk, Ross. Akad. Nauk 408, No. 5, 591-595 (2006).
From the text: 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. In this paper, we solve the problem of homotopy classification on an arbitrary stratified manifold. The classification is in terms of the $$K$$-homology of a stratified manifold. The ideas of Atiyah concerning the realization of $$K$$-homology by generalized elliptic operators [M. F. Atiyah, in: Proc. Int. Conf. Funct. Anal. Rel. Topics, Tokyo 1969, 21–30 (1970; Zbl 0193.43601)] allow us to avoid the difficulties caused by the fact that, in the situation under consideration, the operators modulo compact operators are determined by whole sets of symbols on strata rather than by only one symbol, and that the ellipticity condition is of an infinite-dimensional character. As applications, we obtain an index formula and a topological obstruction for Fredholm problems and calculate the $$K$$-group of algebras of pseudodifferential operators (PDOs).

##### MSC:
 58J20 Index theory and related fixed-point theorems on manifolds 58J40 Pseudodifferential and Fourier integral operators on manifolds 19K33 Ext and $$K$$-homology 35S05 Pseudodifferential operators as generalizations of partial differential operators 46L80 $$K$$-theory and operator algebras (including cyclic theory) 47F05 General theory of partial differential operators (should also be assigned at least one other classification number in Section 47-XX)
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##### References:
 [1] M. F. Atiyah, in Proceedings of the International Symposium on Functional Analysis (Tokyo, 1969), pp. 21–30. [2] M. S. Agranovich and M. I. Vishik, Usp. Mat. Nauk 19(3), 53–161 (1964). [3] B. A. Plamenevskii and V. N. Senichkin, Algebra Analiz 13(6), 124–174 (2001). [4] V. P. Maslov, Perturbation Theory and Asymptotic Methods (Mosk. Gos. Univ., Moscow, 1965) [in Russian]. [5] A. Connes and N. Higson, C. R. Acad. Sci., Sér. I 311(2), 101–106 (1990). [6] N. Higson and J. Roe, Analytic K-Homology (Oxford Univ. Press, Oxford, 2000). [7] G. Luke, J. Differ. Equations 12, 566–589 (1972). · Zbl 0238.35077 · doi:10.1016/0022-0396(72)90026-5
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