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On the homotopy classification of elliptic operators on manifolds with corners. (English. Russian original) Zbl 1156.58011
Dokl. Math. 75, No. 2, 186-189 (2007); translation from Dokl. Akad. Nauk, Ross. Akad. Nauk 413, No. 1, 16-19 (2007).
From the introduction: This paper deals with elliptic theory on manifolds with corners. Such manifolds arise, e.g., as products of manifolds with boundary. We determine a natural dual object for a manifold with corners and show how a calculus of zeroth-order pseudodifferential operators (PDOs) on such a manifold can be constructed in terms of the localization principle on \(C^*\)-algebras. The definition uses induction on the “depth” of manifolds and has a natural formulation for operators with a parameter. The results are applied to solve Gelfand’s problem about a homotopy classification of elliptic operators for manifolds with corners. One of the consequences of the classification is an explicit formula for Atiyah-Bott-type obstructions to setting Fredholm problems.

58J40 Pseudodifferential and Fourier integral operators on manifolds
58J05 Elliptic equations on manifolds, general theory
Full Text: DOI
[1] R. Melrose, in Proceedings of the International Congress of Mathematicians, Kyoto (Heidelberg, Berlin, 1990), pp. 217–234.
[2] B. Monthubert, J. Funct. Anal. 199(1), 243–286 (2003). · Zbl 1025.58009 · doi:10.1016/S0022-1236(02)00038-1
[3] B. A. Plamenevskii and V. N. Senichkin, Algebra Analiz 13(6), 124–174 (2001).
[4] V. P. Maslov, Operator Methods (Nauka, Moscow, 1973) [in Russian].
[5] A. Savin, K-Theory 34(1), 71–98 (2005). · Zbl 1087.58013 · doi:10.1007/s10977-005-1515-1
[6] V. E. Nazaikinskii, A. Yu. Savin, and B. Yu. Sternin, arXiv:math.KT/0608332.
[7] N. Higson and J. Roe, Analytic K-Homology (Oxford Univ. Press, Oxford, 2000).
[8] B. Monthubert and V. Nistor, arXiv:math.KT/0507601.
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