Secondary cohomology operations and the loop space cohomology

TL;DR

This paper develops secondary cohomology operations to analyze the Hopf algebra structure of loop space cohomology, providing explicit calculations and applications to the exceptional group F4.

Contribution

It constructs secondary cohomology operations as a Hopf algebra for simply connected spaces and computes the loop space cohomology ring explicitly.

Findings

Loop space cohomology ring is described by generators and relations.

Secondary operations determine the decomposition of the Hopf algebra.

Application to the exceptional group F4's 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.