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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\textit{V. E. Nazaikinskii} et al., Dokl. Math. 73, No. 3, 407--411 (2006; Zbl 1327.58025); translation from Dokl. Akad. Nauk, Ross. Akad. Nauk 408, No. 5, 591--595 (2006)

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##### References:

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[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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