×

A generalization of manifold and its characteristic classes. (English. Russian original) Zbl 0714.57012

Funct. Anal. Appl. 24, No. 1, 26-32 (1990); translation from Funkts. Anal. Prilozh. 24, No. 1, 29-37 (1990).
The notion of an \(R_ n\)-set has been introduced by the author in Funkts. Anal. Prilozh. 21, No.3, 38-52 (1987; Zbl 0632.57016). The structures of \(R_ n\)-sets may be defined on the sets of transitivity classes of local diffeomorphisms on n-manifolds and on the sets of leaves of foliations of codimension n. The characteristic classes of \(R_ n\)- sets are obtained by using the cohomologies \(H^*(W_ n)\), \(H^*(W_ n,GL_ n)\), \(H^*(W_ n,O(n))\), where \(W_ n\) is the Lie algebra of formal vector fields on \({\mathbb{R}}^ n\), \(GL_ n=GL(n,R)\), \(O(n)=the\) orthogonal group. The Pontryagin classes and the classes induced by the characteristic classes of foliations may be found among these characteristic classes. The Pontryagin classes are real but are no longer integer valued. The introduced characteristic classes are not trivial in the case of \(H^*(W_ 1,GL_ 1)\).
Reviewer: V.Oproiu

MSC:

57R20 Characteristic classes and numbers in differential topology
57R30 Foliations in differential topology; geometric theory
58H05 Pseudogroups and differentiable groupoids
58A99 General theory of differentiable manifolds
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] I. N. Bernshtein and B. I. Rozenfel’d, ”Homogeneous spaces of infinite-dimensional Lie algebras and characteristic classes of foliations,” Usp. Mat. Nauk.,28, No. 4, 103-138 (1973).
[2] V. V. Vagner, ”Theory of differential objects and foundations of differential geometry,” in: O. Veblen and J. Whitehead, Foundations of Differential Geometry [Russian translation], IL, Moscow (1949). · Zbl 0041.29603
[3] I. M. Gel’fand and D. B. Fuks, ”Cohomology of the Lie algebra of formal vector fields,” Dokl. Akad. Nauk SSSR, Ser. Mat.,34, No. 2, 322-337 (1970).
[4] M. V. Losik, ”Characteristic classes of structures on manifolds,” Funkts. Anal. Prilozhen.,21, No. 3, 38-52 (1987). · Zbl 0632.57016
[5] R. O. Wells, Jr., Differential Analysis on Complex Manifolds, Prentice-Hall (1973). · Zbl 0262.32005
[6] A. Haefliger, ”Feuilletages sur les vari?t?s ouvertes,” Topology,9, No. 2, 183-194 (1970). · Zbl 0196.26901 · doi:10.1016/0040-9383(70)90040-6
[7] J. Pradines, ”Vari?t?s d’orbites,” Publ. D?p. Math.,63, No. 2A, 65-70 (1987). · Zbl 0619.57008
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.