Characteristic classes of transformation groups
For a transformation group \(G\) of a topological space \(X\) a spectral sequence with the term \(E_ 2^{p,q} = H^ p(G; H^ q(X;A))\), where \(A\) is a coefficient group, is introduced. The cohomology of the complex of \(G\)-invariant cochains with coefficients in \(A\) determines the characteristic classes of \(G\) as elements of \(E_ \infty\): in particular cases they are elements of \(H^ p(G;H^ q(X;A))\) (or \(H^ p_ c(G; H^ q(X;A)\)) for a continuous transformation group \(G\)). The main applications concern the case when \(A = \mathbb{R}\) and the de Rham complex is used. It is shown that the characteristic classes of \textit{A. G. Rejman, M. A. Semenov-Tyan-Shanskij} and \textit{L. D. Faddeev} [Funct. Anal. Appl. 18, 319-326 (1984); translation from Funkts. Anal. Prilozh. 18, No. 4, 64-72 (1984; Zbl 0588.58001)] for the automorphism group of a smooth principal fibre bundle and the characteristic classes of \textit{R. Bott} [Enseign. Math., II. Ser. 23, 209-220 (1977; Zbl 0367.57004)] for the diffeomorphism group of a manifold are the partial cases of the above construction. The connection of the above characteristic classes with the functional of \textit{M. F. Atiyah, V. K. Patodi} and \textit{I. M. Singer} [Math. Proc. Camb. Phil. Soc. 77, 43-69 (1975; Zbl 0297.58008), 78, 405- 432 (1975; Zbl 0314.58016)] is indicated.
automorphism group of a smooth principal fibre bundle
scientific article; zbMATH DE number 444847