Sullivan conjecture

In mathematics, Sullivan conjecture or Sullivan's conjecture on maps from classifying spaces can refer to any of several results and conjectures prompted by homotopy theory work of Dennis Sullivan. A basic theme and motivation concerns the fixed point set in group actions of a finite group formula_1. The most elementary formulation, however, is in terms of the classifying space formula_2 of such a group. Roughly speaking, it is difficult to map such a space formula_2 continuously into a finite CW complex formula_4 in a non-trivial manner. Such a version of the Sullivan conjecture was first proved by Haynes Miller. Specifically, in 1984, Miller proved that the function space, carrying the compact-open topology, of base point-preserving mappings from formula_2 to formula_4 is weakly contractible.
This is equivalent to the statement that the map formula_4 → formula_8 from X to the function space of maps formula_2 → formula_4, not necessarily preserving the base point, given by sending a point formula_11 of formula_4 to the constant map whose image is formula_11 is a weak equivalence. The mapping space formula_8 is an example of a homotopy fixed point set. Specifically, formula_8 is the homotopy fixed point set of the group formula_1 acting by the trivial action on formula_4. In general, for a group formula_1 acting on a space formula_4, the homotopy fixed points are the fixed points formula_20 of the mapping space formula_21 of maps from the universal cover formula_22 of formula_2 to formula_4 under the formula_1-action on formula_21 given by formula_27 in formula_1 acts on a map formula_29 in formula_21 by sending it to formula_31. The formula_1-equivariant map from formula_22 to a single point formula_34 induces a natural map η: formula_35→formula_20 from the fixed points to the homotopy fixed points of formula_1 acting on formula_4. Miller's theorem is that η is a weak equivalence for trivial formula_1-actions on finite-dimensional CW complexes. An important ingredient and motivation (see [1]) for his proof is a result of Gunnar Carlsson on the homology of formula_40 as an unstable module over the Steenrod algebra.
Miller's theorem generalizes to a version of Sullivan's conjecture in which the action on formula_4 is allowed to be non-trivial. In, Sullivan conjectured that η is a weak equivalence after a certain p-completion procedure due to A. Bousfield and D. Kan for the group formula_42. This conjecture was incorrect as stated, but a correct version was given by Miller, and proven independently by Dwyer-Miller-Neisendorfer, Carlsson, and Jean Lannes, showing that the natural map formula_43 → formula_44 is a weak equivalence when the order of formula_1 is a power of a prime p, and where formula_46 denotes the Bousfield-Kan p-completion of formula_4. Miller's proof involves an unstable Adams spectral sequence, Carlsson's proof uses his affirmative solution of the Segal conjecture and also provides information about the homotopy fixed points formula_48 before completion, and Lannes's proof involves his T-functor.