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 G {\displaystyle G} . The most elementary formulation, however, is in terms of the classifying space B G {\displaystyle BG} of such a group. Roughly speaking, it is difficult to map such a space B G {\displaystyle BG} continuously into a finite CW complex X {\displaystyle X} 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 B G {\displaystyle BG} to X {\displaystyle X} is weakly contractible. This is equivalent to the statement that the map X {\displaystyle X} → F ( B G , X ) {\displaystyle F(BG,X)} from X to the function space of maps B G {\displaystyle BG} → X {\displaystyle X} , not necessarily preserving the base point, given by sending a point x {\displaystyle x} of X {\displaystyle X} to the constant map whose image is x {\displaystyle x} is a weak equivalence. The mapping space F ( B G , X ) {\displaystyle F(BG,X)} is an example of a homotopy fixed point set. Specifically, F ( B G , X ) {\displaystyle F(BG,X)} is the homotopy fixed point set of the group G {\displaystyle G} acting by the trivial action on X {\displaystyle X} . In general, for a group G {\displaystyle G} acting on a space X {\displaystyle X} , the homotopy fixed points are the fixed points F ( E G , X ) G {\displaystyle F(EG,X)^{G}} of the mapping space F ( E G , X ) {\displaystyle F(EG,X)} of maps from the universal cover E G {\displaystyle EG} of B G {\displaystyle BG} to X {\displaystyle X} under the G {\displaystyle G} -action on F ( E G , X ) {\displaystyle F(EG,X)} given by g {\displaystyle g} in G {\displaystyle G} acts on a map f {\displaystyle f} in F ( E G , X ) {\displaystyle F(EG,X)} by sending it to g f g − 1 {\displaystyle gfg^{-1}} . The G {\displaystyle G} -equivariant map from E G {\displaystyle EG} to a single point ∗ {\displaystyle *} induces a natural map η: X G = F ( ∗ , X ) G {\displaystyle X^{G}=F(*,X)^{G}} → F ( E G , X ) G {\displaystyle F(EG,X)^{G}} from the fixed points to the homotopy fixed points of G {\displaystyle G} acting on X {\displaystyle X} . Miller's theorem is that η is a weak equivalence for trivial G {\displaystyle G} -actions on finite-dimensional CW complexes. An important ingredient and motivation for his proof is a result of Gunnar Carlsson on the homology of B Z / 2 {\displaystyle BZ/2} as an unstable module over the Steenrod algebra. Miller's theorem generalizes to a version of Sullivan's conjecture in which the action on X {\displaystyle X} is allowed to be non-trivial. In 1971, Sullivan conjectured that η is a weak equivalence after a certain p-completion procedure due to A. Bousfield and D. Kan for the group G = Z / 2 {\displaystyle G=Z/2} . 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 ( X G ) p {\displaystyle (X^{G})_{p}} → F ( E G , ( X ) p ) G {\displaystyle F(EG,(X)_{p})^{G}} is a weak equivalence when the order of G {\displaystyle G} is a power of a prime p, and where ( X ) p {\displaystyle (X)_{p}} denotes the Bousfield-Kan p-completion of X {\displaystyle X} . 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 F ( E G , X ) G {\displaystyle F(EG,X)^{G}} before completion, and Lannes's proof involves his T-functor.
References
External links Gottlieb, Daniel H. (2001) [1994], "Sullivan conjecture", Encyclopedia of Mathematics, EMS Press Book extract J. Lurie's course notes