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

Harris, John C.; Shank, R. James

doi:10.1090/S0002-9947-1992-1118825-6

Lannes’ $T$ functor on summands of $H^ *(B(\textbf {Z}/p)^ s)$
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by John C. Harris and R. James Shank PDF

Trans. Amer. Math. Soc. 333 (1992), 579-606 Request permission

Abstract:

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

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