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 CMFD, 2008, Volume 29, Pages 131–164 (Mi cmfd127)

Noncommutative geometry and classification of elliptic operators

V. E. Nazaikinskiia, A. Yu. Savinbc, B. Yu. Sterninbc

a A. Ishlinsky Institite for Problems in Mechanics, Russian Academy of Sciences
b Independent University of Moscow
c Russian State Social University

Abstract: The computation of a stable homotopic classification of elliptic operators is an important problem of elliptic theory. The classical solution of this problem is given by Atiyah and Singer for the case of smooth compact manifolds. It is formulated in terms of $K$-theory for a cotangent fibering of the given manifold. It cannot be extended for the case of nonsmooth manifolds because their cotangent fiberings do not contain all necessary information. Another Atiyah definition might fit in such a case: it is based on the concept of abstract elliptic operators and is given in term of $K$-homologies of the manifold itself (instead of its fiberings). Indeed, this theorem is recently extended for manifolds with conic singularities, ribs, and general so-called stratified manifolds: it suffices just to replace the phrase “smooth manifold” by the phrase “stratified manifold” (of the corresponding class). Thus, stratified manifolds is a strange phenomenon in a way: the algebra of symbols of differential (pseudodifferential) operators is quite noncommutative on such manifolds (the symbol components corresponding to strata of positive codimensions are operator-valued functions), but the solution of the classification problem can be found in purely geometric terms. In general, it is impossible for other classes of nonsmooth manifolds. In particular, the authors recently found that, for manifolds with angles, the classification is given by a $K$-group of a noncommutative $C^*$-algebra and it cannot be reduced to a commutative algebra if normal fiberings of faces of the considered manifold are nontrivial. Note that the proofs are based on noncommutative geometry (more exactly, the K-theory of $C^*$-algebras) even in the case of stratified manifolds though the results are “classical.” In this paper, we provide a review of the abovementioned classification results for elliptic operators on manifolds with singularities and corresponding methods of noncommutative geometry (in particular, the localization principle in $C^*$-algebras).

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English version:
Journal of Mathematical Sciences, 2010, 164:4, 603–636

Bibliographic databases:

UDC: 515.168.5+517.986.32

Citation: V. E. Nazaikinskii, A. Yu. Savin, B. Yu. Sternin, “Noncommutative geometry and classification of elliptic operators”, Proceedings of the Crimean autumn mathematical school-symposium, CMFD, 29, PFUR, M., 2008, 131–164; Journal of Mathematical Sciences, 164:4 (2010), 603–636

Citation in format AMSBIB
\Bibitem{NazSavSte08} \by V.~E.~Nazaikinskii, A.~Yu.~Savin, B.~Yu.~Sternin \paper Noncommutative geometry and classification of elliptic operators \inbook Proceedings of the Crimean autumn mathematical school-symposium \serial CMFD \yr 2008 \vol 29 \pages 131--164 \publ PFUR \publaddr M. \mathnet{http://mi.mathnet.ru/cmfd127} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=2472267} \elib{http://elibrary.ru/item.asp?id=15317220} \transl \jour Journal of Mathematical Sciences \yr 2010 \vol 164 \issue 4 \pages 603--636 \crossref{https://doi.org/10.1007/s10958-010-9765-8} \scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-77949296336} 

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This publication is cited in the following articles:
1. Yu. A. Kordyukov, V. A. Pavlenko, “Singular integral operators on a manifold with a distinguished submanifold”, Ufa Math. J., 6:3 (2014), 35–68
2. A. Yu. Savin, B. Yu. Sternin, “Homotopy classification of elliptic problems associated with discrete group actions on manifolds with boundary”, Ufa Math. J., 8:3 (2016), 122–129
3. A. Yu. Savin, “O gomotopicheskoi klassifikatsii ellipticheskikh zadach so szhatiyami i $K$-gruppakh sootvetstvuyuschikh $C^*$-algebr”, Differentsialnye i funktsionalno-differentsialnye uravneniya, SMFN, 64, no. 1, Rossiiskii universitet druzhby narodov, M., 2018, 164–179
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