Secondary cohomology operations and the loop space cohomology

Abstract

Motivated by the loop space cohomology we construct the secondary operations on the cohomology $H^*(X; \mathbb{Z}_p)$ to be a Hopf algebra for a simply connected space $X.$ The loop space cohomology ring $H^*(\Omega X; \mathbb{Z}_p)$ is calculated in terms of generators and relations. This answers to A. Borel's decomposition of a Hopf algebra into a tensor product of the monogenic ones in which the heights of generators are determined by means of the action of the primary and secondary cohomology operations on $H^*(X;\mathbb{Z}_p).$ An application for calculating of the loop space cohomology of the exceptional group $F_4$ is given.