The loop cohomology of a space with the polynomial cohomology algebra

TL;DR

This paper computes the loop cohomology algebra of certain simply connected spaces with polynomial cohomology, revealing conditions under which it forms an exterior algebra based on Steenrod operations.

Contribution

It provides a method to determine the structure of loop cohomology algebra using Steenrod operations, extending Borel's theorem with a new converse condition.

Findings

Loop cohomology algebra is exterior if and only if $Sq_1$ is multiplicatively decomposable.

Calculated $H^*(\, ext{loop space}\, X)$ for spaces with polynomial cohomology.

Established a criterion linking Steenrod operations to algebraic structure of loop cohomology.

Abstract

Given a simply connected space $X$ with the cohomology $H^*(X;{\mathbb Z}_2)$ to be polynomial, we calculate the loop cohomology algebra $H^*(\Omega X;{\mathbb Z}_2)$ by means of the action of the Steenrod cohomology operation $Sq_1$ on $H^*(X;{\mathbb Z}_2).$ As a consequence we obtain that $H^*(\Omega X;{\mathbb Z}_2)$ is the exterior algebra if and only if $Sq_1$ is multiplicatively decomposable on $H^{\ast}(X;{\mathbb Z}_2).$ The last statement in fact contains a converse of a theorem of A. Borel.