Differential graded modules over a nonconnected differential graded algebra
Let \(\Lambda_*\) be an associative differential graded algebras over a hereditary ring k. To each differential graded \(\Lambda_*\)-module \(M_*\) the author associates ``characteristic'' classes \(\theta_ q(M_*)\in Ext^ 2_{\Lambda}(H_ q(M_*),H_{q+1}(M_*))\) which depend only on the quasi-isomorphism class of \(M_*.\) If \(C_*\) is a dg-coalgebra and if t: \(C_*\to \Lambda_*\) is a twisting cochain, then there is a spectral sequence with \(E^ 2=H_ p(C_*;H_ q(M_*))\) converging to \(H_*(C_*\otimes_ tM_*)\). The author proves a formula (generalizing a formula of Hurewicz-Fadell) which expresses the \(d^ 2\)-differential in terms of the characteristic classes \(\theta_ q(M_*)\) and a class \([t^ 2]\in H^ 2(C_*;H_ 1(\Lambda_*))\) which is related to t. In particular, the author gives a formula (involving Stiefel-Whitney- classes) of the \(d^ 2\)-differential of the Serre spectral sequence of an O(n)-bundle \(X\to E\to B\).
scientific article; zbMATH DE number 4205404
scientific article