×

zbMATH — the first resource for mathematics

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.

MSC:
58J40 Pseudodifferential and Fourier integral operators on manifolds
58J05 Elliptic equations on manifolds, general theory
PDF BibTeX XML Cite
Full Text: DOI
References:
[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.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.