 
Summary: The functor T and the cohomology
of mapping spaces
Jaume Aguad´e, Carles Broto, and Laia Saumell
1. Introduction
In his fundamental work [15] Lannes has introduced a functor T defined in the
category K (resp. U) of unstable algebras (resp. modules) over the Steenrod algebra
which has many important applications in homotopy theory. This functor is, in
some sense, the algebraic analogue of the mapping space functor Map(BV, ) for
an elementary abelian group V . More precisely, there is a functor T = TV : K K
which is adjoint to the functor  H
(BV ): K K, and which computes, under
some hypothesis, H
(Map(BV, 
p )). Here, and throughout this paper, H
()
denotes H
(; Fp) for some fixed prime p and Y
p denotes the pcompletion of
Y in the sense of BousfieldKan ([7]). We also assume that all constructions are
done simplicially. For practical purposes, one would like to work on a component
