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.
Categories: math.AT