The loop cohomology of a space with the polynomial cohomology algebra

Abstract

Given a simply connected space $X$ with polynomial cohomology $H^{\ast }(X;Z_2),$ we calculate the loop cohomology algebra $H^{\ast }(\Omega X;Z_2)$ by means of the action of the Steenrod cohomology operation $S{q}_1$ on $H^{\ast }(X;Z_2).$ This calculation uses an explicit construction of the minimal Hirsch filtered model of the cochain algebra $C^{\ast }(X;Z_2).$ As a consequence we obtain that $H^{\ast }(\Omega X;Z_2)$ is the exterior algebra if and only if $S{q}_1$ is multiplicatively decomposable on $H^{\ast }(X;Z_2).$ The last statement in fact contains a converse of a theorem of A. Borel (Switzer, 1975, Theorem 15.60).