Scalar differential invariants and characteristic classes of homogeneous geometrical structures
It was pointed out in [\textit{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 \(H^{\ell + m + s} (\text{BSO}(\ell,m))\), \(s > 0\).
cohomology of regular \(\mathbb{R}\)-spectrum
algebra of scalar differential invariants
scientific article; zbMATH DE number 525688