Steenrod algebra module maps from 𝐻*(𝐵(𝑍/𝑝)ⁿ) to 𝐻*(𝐵(𝑍/𝑝)^{𝑠})

Abstract

Let H ⊗ n {H^{ \otimes n}} denote the mod − p \operatorname {mod}-p cohomology of the classifying space B ( Z / p ) n B{({\mathbf {Z}}/p)^n} as a module over the Steenrod algebra A \mathcal {A} . Adams, Gunawardena, and Miller have shown that the n × s n \times s matrices with entries in Z / p {\mathbf {Z}}/p give a basis for the space of maps Ho m A ( H ⊗ n , H ⊗ s ) {\text {Ho}}{{\text {m}}_\mathcal {A}}({H^{ \otimes n}},{H^{ \otimes s}}) . For n n and s s relatively prime, we give a new basis for this space of maps using recent results of Campbell and Selick. The main advantage of this new basis is its compatibility with Campbell and Selick’s direct sum decomposition of H ⊗ n {H^{ \otimes n}} into ( p n − 1 ) ({p^n} - 1) A \mathcal {A} -modules. Our applications are at the prime two. We describe the unique map from H ¯ \bar H to D ( n ) D(n) , the algebra of Dickson invariants in H ⊗ n {H^{ \otimes n}} , and we give the dimensions of the space of maps between the indecomposable summands of H ⊗ 3 {H^{ \otimes 3}} .