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
The most elementary formulation, however, is in terms of the classifying space
Roughly speaking, it is difficult to map such a space
continuously into a finite CW complex
Such a version of the Sullivan conjecture was first proved by Haynes Miller.
[1] Specifically, in 1984, Miller proved that the function space, carrying the compact-open topology, of base point-preserving mappings from
This is equivalent to the statement that the map
from X to the function space of maps
, not necessarily preserving the base point, given by sending a point
to the constant map whose image is
is an example of a homotopy fixed point set.
is the homotopy fixed point set of the group
acting by the trivial action on
acting on a space
, the homotopy fixed points are the fixed points
of maps from the universal cover
induces a natural map η:
from the fixed points to the homotopy fixed points of
Miller's theorem is that η is a weak equivalence for trivial
-actions on finite-dimensional CW complexes.
An important ingredient and motivation for his proof is a result of Gunnar Carlsson on the homology of
as an unstable module over the Steenrod algebra.
[2] Miller's theorem generalizes to a version of Sullivan's conjecture in which the action on
In,[3] Sullivan conjectured that η is a weak equivalence after a certain p-completion procedure due to A. Bousfield and D. Kan for the group
This conjecture was incorrect as stated, but a correct version was given by Miller, and proven independently by Dwyer-Miller-Neisendorfer,[4] Carlsson,[5] and Jean Lannes,[6] showing that the natural map
is a weak equivalence when the order of
denotes the Bousfield-Kan p-completion of
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
before completion, and Lannes's proof involves his T-functor.