Secondary cohomology and the Steenrod square

We introduce and study various properties of the secondary cohomology of a space. Certain Steenrod squares are shown to be related to the action of the symmetric groups on secondary cohomology. To Jan–Erik Roos on his sixty–Þfth birthday For a Þeld k we choose the Eilenberg–MacLane space Z n = K(k;n) by the realization of the simplicial k–vector space generated by the non–basepoint singular simplices of the n–sphere S n = S 1 ^:::^S 1 . The permutation of the smash product factors S 1 yields an action of the symmetric group on on S n and hence on Z n . Moreover the quotient map S n C S m ! S n+m induces a cup product map n : Z n C Z m ! Z m+n with n;m > 1; see the Appendix below. It is well known that the (reduced) cohomology e H n (X;k) of a path-connected pointed space X is the same as the set [X;Z n ] of homotopy classes fxg of pointed maps x : X ! Z n . Moreover the cup product of the cohomology algebra H E = H E (X;k) = e H E a k is induced by the map n, that is fxg [ fyg = fn(x;y)g. The