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Transactions of the American Mathematical Society

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Lannes' $ T$ functor on summands of $ H\sp *(B({\bf Z}/p)\sp s)$

Authors: John C. Harris and R. James Shank
Journal: Trans. Amer. Math. Soc. 333 (1992), 579-606
MSC: Primary 55S10
MathSciNet review: 1118825
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Abstract: Let $ H$ be the $ \bmod$-$ 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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