Cohomology operations derived from modular invariants (Q1079231)

This paper is the continuation of the author's earlier paper [Topology theory and applications, 5th Colloq., Eger/Hung. 1983, Colloq. Math. Soc. János Bolyai 41, 345-355 (1985)]. The purpose is to demonstrate a similar result for \({\mathcal A}(p)\), for \(p>2\), by using a free subalgebra of invariants of the linear subgroup \(SL_ n\subset GL_ n\), which operates naturally on the cohomology algebra \(H^*(E^ n)\) of an elementary abelian p-group \(E^ n\). Let K be a chain complex with diagonal \(K\to K\otimes K\) and \(K^ V=\otimes_{x\in V}K_ x\), \(K_ x=K\). Then we have the map \[ H^ q(K)\to_{P_{E^ n}}H_{E^ n}^{p^{n_ q}}(W(E^ n)\otimes K^ V)\to_{d^*_{E^ n}}H^*(E^ n)\otimes H^*(K), \] where \(P_{E^ n}\) is the Steenrod power map, \(d^*_{E^ n}\) is induced by the diagonal \(K\to K^ V\) and the Künneth isomorphism. Let \(L_ n\), \(Q_{n,s}\) be the Dickson invariants. Then he shows: For every (S,R) with \(R=(r_ 1,r_ 2,....,r_ n)\), \(r_ i\geq 0\), and \(S=(s_ 1,...,s_ k)\), \(0\leq s_ 1<...<s_ k\leq n-1\), there exists a stable cohomology operation \(St^{S,R}\) such that \[ d^*_{E^ n}P_{E^ n}u=\mu (q)^ n\sum (-1)^{r(S,R)} \tilde M_{n,s_ 1}...M\tilde {\;}_{n,s_ k} \tilde L_ n^{r_ 0} Q^{r_ 1}_{n,1}...Q^{r_{n-1}}_{\quad n,n- 1}\otimes St^{S,R_ u} \] for \(u\in H^ q(K)\). Here \(r_ 0=q-k- 2(r_ 1+...+r_ n)\) and \(r(S,R)=k+s_ 1+...+s_ k+r_ 1+2r_ 2+...+nr_ n\). The main result is \(St^{S,R}=(\tau_ S\xi^ R)^*\), which denotes the dual of the basis \(\tau_ S\xi^ R=\tau_{s_ 1}...\tau_{s_ k}\xi_ 1^{r_ 1}...\xi_ n^{r_ n}\) of the algebra \({\mathcal A}(p)^*=E(\tau_ 0,\tau_ 1,...)\otimes P(\xi_ 1,\xi_ 2,...)\).