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Is there any reference to the proof of following: let $T$ denote the Lannes functor. Then (see the link above for more details) for any finite $E$-complex $X$ (where $E$ is finite-dimensional $\mathbb F_p$-vector space), one should have $T_EH_E^*(X) = H^*BE \otimes H^*(X^E)$?

Wilkerson and Dwyer (”Smith theory and the functor T”, p. 2) give a reference to the unpublished manuscript “Cohomology of groups and function spaces” by Lannes. But I can not find it anywhere.

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    $\begingroup$ Generally speaking, questions should be as self-contained as possible, so it would be better to give at least a hint of the notation than to defer entirely to an external article. $\endgroup$
    – LSpice
    Commented Sep 12, 2022 at 22:05
  • $\begingroup$ @LSpice, I have added some details. $\endgroup$
    – Vanya Karpov
    Commented Sep 12, 2022 at 22:09
  • $\begingroup$ @LSpice, and even more details. $\endgroup$
    – Vanya Karpov
    Commented Sep 12, 2022 at 22:16

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There is something wrong with your assertion, which, when $X$ is specialized to a point, says that $$T_EH^*(BE) = H^*(BE).$$ But this is not true. (I can't access the Dywer-Wilkerson paper you mention right at the moment.)

Lannes' famous 1992 paper in Pub. IHES (en francais!) has a functor he calls $Fix$, and $Fix(H_E^*(X)) \simeq H^*(X^E)$ for $X$ a finite $E$--complex. $Fix$ is a variant of $T_E$. Maybe this will help you.

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