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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})      .

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