Lannes’ 𝑇 functor on summands of 𝐻*(𝐵(𝑍/𝑝)^{𝑠})

Abstract

Let H H be the mod - p \bmod \text {-}p cohomology of the classifying space B ( Z / p ) B({\mathbf {Z}}/p) thought of as an object in the category, U \mathcal {U} , of unstable modules over the Steenrod algebra. Lannes constructed a functor T : U → U T:\mathcal {U} \to \mathcal {U} which is left adjoint to the functor A ↦ A ⊗ H A \mapsto A \otimes H . In this paper we evaluate T T on the indecomposable U \mathcal {U} -summands of H ⊗ s {H^{ \otimes s}} , the tensor product of s s copies of H H . Our formula involves the composition factors of certain tensor products of irreducible representations of the semigroup ring F p [ M s , s ( Z / p ) ] {{\mathbf {F}}_p}[{{\mathbf {M}}_{s,}}_s({\mathbf {Z}}/p)] . The main application is to determine the homotopy type of the space of maps from B ( Z / p ) ] B({\mathbf {Z}}/p)] to X X when X X is a wedge summand of the space Σ ( B ( Z / p ) s ) \Sigma (B{({\mathbf {Z}}/p)^s}) .