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 $T_EH_E^*(X) = H^*BE \otimes H^*(X^E)$?

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Wilkerson and Dwyer (”Smith theory and the functor T”, p. 2) give a reference to the unpublished manuscript “Cohomology of groups and function spaces” by Lannes.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.

Is there any reference to the proof of following: let $T$ denote the Lannes functor. Then (see the link above for the notation) $T_EH_E^*(X) = H^*BE \otimes H^*(X^E)$?

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 $T_EH_E^*(X) = H^*BE \otimes H^*(X^E)$?