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

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

Zbl 0735.57012