Is there any reference to the proof of following: let $T$ denote the [Lannes functor.](https://encyclopediaofmath.org/wiki/Lannes-T-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Wilkerson and Dwyer (”Smith theory and the functor T”, p. 2) give a reference to the unpublished manuscript “Cohomology of groups and function spaces” by Lannes.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.



Fixed points cohomology via Lannes T-functor
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.


