The coalgebra structure of the cohomology of multiplicative fibrations (Q387218)

If \(f: X\to K\) is an \(H\)-map, with \(X\) simply connected and \(K\) a generalized Eilenberg-MacLane space, the fibre space, \(\tilde{X}\), inherits a structure of \(H\)-space and the fibration \(\rho: \tilde{X}\to K\), of fibre \(j: \Omega X\to \tilde{X}\), is multiplicative. When \(X\) is a generalized Eilenberg-MacLane space and \(f\) a loop map, the coalgebra structure of \(H^*(\tilde{X};{\mathbb Z}_{p})\) has been determined by \textit{J. Harper} and \textit{C. Schochet} [Math. Scand. 29, 232--236 (1972; Zbl 0235.55019)]. The general case was considered also by \textit{R. M. Kane} [Mem. Am. Math. Soc. 340, 110 p. (1986; Zbl 0585.57024)].NEWLINENEWLINEIn the paper under review, the author strengthens the known results and proves the existence of a choice of algebra generators of \(H^*(\tilde{X};{\mathbb Z}_{p})\) with reduced coproduct in \(\mathrm{Im}\,\rho^*\otimes \mathrm{Im}\,\rho^*\). The proof relies on a Borel decomposition of \(H^*(\tilde{X};{\mathbb Z}_{p})\) arising from the analysis of the Serre spectral sequence of \(\rho:\tilde{X}{\rightarrow} X\). As a consequence of the main result, if \(H^*(\tilde{X};{\mathbb Z}_{p})\) is an associative ring, then \(\mathrm{Im}\,j_{*}\) is a central subalgebra of \(H_{*}(\tilde{X};{\mathbb Z}_{p})\).