Scalar differential invariants and characteristic classes of homogeneous geometrical structures. (English. Russian original) Zbl 0814.57019

Verbovetskij, A. M.; Vinogradov, A. M.; Gessler, D. M.

Math. Notes 51, No. 6, 543-549 (1992); translation from Mat. Zametki 51, No. 6, 15-26 (1992).

It was pointed out in [A. M. Vinogradov, Scalar differential invariants, diffieties and characteristic classes, in: ‘Mechanics, analysis and geometry: 200 years after Lagrange, 379-414 (1991; Zbl 0735.57012)], the relationship between the algebra of scalar differential invariants of homogeneous geometrical structures and their characteristic classes, namely, that the characteristic classes are the cohomology classes of the regular \(\mathbb{R}\)-spectrum of the corresponding algebra of differential invariants.
In the present paper the next step is taken up and the cohomology of the above-mentioned regular \(\mathbb{R}\)-spectrum is computed. Namely, it is shown that this cohomology coincides with the cohomology of the classifying space BG of the subgroup \(G\) of the general differential group \(\mathbb{G}^ p(n)\) that defines the relevant geometrical structure of order \(p\). For example, the characteristic classes of pseudo-Riemannian metrics of type \((\ell,m)\) are exhausted by the cohomology classes \(H^ i(\text{BSO}(\ell,m))\), \(0\leq i\leq \ell + m\), and the characteristic classes of their \(s\)-dimensional bordisms by the cohomology classes \(H^{\ell + m + s} (\text{BSO}(\ell,m))\), \(s > 0\).

Reviewer: V.L.Popov (Moskva)

MSC:

57R15	Specialized structures on manifolds (spin manifolds, framed manifolds, etc.)
57R20	Characteristic classes and numbers in differential topology
55R40	Homology of classifying spaces and characteristic classes in algebraic topology

Keywords:

cohomology of regular \(\mathbb{R}\)-spectrum; algebra of scalar differential invariants; homogeneous geometrical structures; characteristic classes; classifying space

Citations:

Zbl 0735.57012