The Eilenberg-Moore spectral sequence and the mod 2 cohomology of certain free loop spaces (Q793411)

The mod 2 cohomology of the space \(\Lambda(X)\) of free loops on X (maps: \(S^ 1\to X)\) is computed for certain spaces X. More precisely: \(\Lambda(X)\) is the homotopy pull back of two copies of the diagonal map \(X\to X\times X,\) thus one can use the Eilenberg-Moore spectral sequence \[ E_ 2=Tor_{H^*(X\times X)}(H^*(X),H^*(X))\Rightarrow H^*(\Lambda(X)). \] Then if \(H^*(X,{\mathbb{Z}}/2{\mathbb{Z}})\) is a truncated polynomial algebra with all truncations at heights which are powers of 2 and \(Sq^ 1H^*(X,{\mathbb{Z}}/2{\mathbb{Z}})=0\) then the above spectral sequence collapses.