The coalgebra structure of the cohomology of multiplicative fibrations

摘要

Let p be an odd prime. Let X be a simply connected H-space and let K be a generalized Eilenberg-MacLane space. Let X-tilde be the fibre space induced from an H-map f : X→K. Then X-tilde inherits an H-structure and there is a multiplicative fibration ΩK→_j X→_p X. The cohomology ring H~*(X-tilde;Z_p) is computed in Lin (2009) [4].In this note, we examine the coalgebra structure of H~*(X-tilde;Z_p). We prove there is a choice of algebra generators for H~*(X-tilde;Z_p) with reduced coproduct in im ρ~* directX im ρ~*.