\(H\)-spaces and the Steenrod algebra (Q2875997)

Let \(p\) be an odd prime and \(X\) a simply connected mod \(p\) finite \(H\)-space with associative homology ring \(H_*(X;\mathbf F_p)\). In the paper under review, the author studies the question whether the algebra generators of a Borel decomposition of \(H^*(X;\mathbf F_p)\) can be chosen to be compatible with the action of the Steenrod algebra \(\mathcal A(p)\). For a Steenrod module \(M\), let \(S(M)\) be the symmetric algebra on \(M\) and \(I\) the ideal in \(S(M)\) generated by elements of the form \(\mathscr P^nx-x^p\) for \(x\in M\) with \(|x|=2n\). The mod \(p\) cohomology of many \(H\)-space has the form \(U(M)=S(M)/I\) for some \(M\).NEWLINENEWLINE A well known counter example is the exceptional Lie group \(E_8\) for \(p=3\) (Example 3).NEWLINENEWLINE Now, let \(\mathcal A(p)^+\) be the subHopf algebra of \(\mathcal A(p)\) generated by \(\mathscr P^{p^k}\) for \(k\geq 0\). Then the author considers the problem from the view point of an \(\mathcal A(p)^+\) algebra.NEWLINENEWLINE For a module \(M\) over \(\mathcal A(p)^+\), put \(U^+(M)=S(M)/I\). Then the author shows that there is an \(\mathcal A(p)^+\) module \(M\) such that \(H^*(X;\mathbf F_p) \cong U^+(M)\) as an \(\mathcal A(p)^+\) algebra. More precisely, let \(B\) be an even degree primitively generated \(\mathcal A(p)\) subHopf algebra of \(H^*(X;\mathbf F_p)\) such that the inclusion \(B \to H^*(X;\mathbf F_p)\) induces an isomorphism \(QB\cong QH^{even}(X;\mathbf F_p)\) on indecomposable modules.NEWLINENEWLINE The existence of such \(B\) is already proved by the author. Let \(R=\{x\in H^*(X;\mathbf F_p)\mid \bar\Delta x\in B \otimes H^*(X;\mathbf F_p)\}\). Then \(R^{even}=B\) and the inclusion \(R^{odd} \to H^*(X;\mathbf F_p)\) induces an isomorphism \(R^{odd} \cong QH^{odd}(X;\mathbf F_p)\). Then the author shows that \(M=\sum PH^{2mp+2}(X;\mathbf F_p)\oplus \xi PH^{2mp+2}(X;\mathbf F_p) \oplus R^{odd}\) is an \(\mathcal A(p)^+\) module, where \(\xi\) is the \(p\)-th power map, and then \(H^*(X;\mathbf F_p) \cong U^+(M)\) as an \(\mathcal A(p)^+\) algebra.