BP-cohomology of mapping spaces from the classifying space of a torus

Let p be a fixed prime number, V a group isomorphic to (Z/p) for some integer d and BV its classifying space. The exceptional properties of the mod p cohomology of BV , as an unstable module and algebra over the Steenrod algebra, have led to the calculation of the mod p cohomology of the mapping spaces with source BV as the image by a functor TV of the mod p cohomology of the target ([La2]). This determination is linked to the solution of the Sullivan conjecture concerning the space of homotopy fixed points for some action of a finite p-group ([Mi]). It is also an essential component of the homotopy theory of Lie groups initiated by Dwyer et Wilkerson ([DW]). We will see that we can deduce from the theory of the T functor (and from its equivariant version) a similar theory relative to the BP-cohomology of the mapping spaces with source the classifying space of a torus when the target is a space whose cohomology with p-adic coefficients is torsion free (p-torsion free space).