Sullivan conjecture

In mathematics, Sullivan conjecture 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 $G$. The most elementary formulation, however, is in terms of the classifying space $BG$ of such a group. Roughly speaking, it is difficult to map such a space $BG$ continuously into a finite CW complex $X$ 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 $BG$ to $X$ is weakly contractible.

This is equivalent to the statement that the map $X$ → $F(BG, X)$ from X to the function space of maps $BG$ → $X$, not necessarily preserving the base point, given by sending a point $x$ of $X$ to the constant map whose image is $x$ is a weak equivalence. The mapping space $F(BG, X)$ is an example of a homotopy fixed point set. Specifically, $F(BG, X)$ is the homotopy fixed point set of the group $G$ acting by the trivial action on $X$. In general, for a group $G$ acting on a space $X$, the homotopy fixed points are the fixed points $F(EG, X)^G$ of the mapping space $F(EG, X)$ of maps from the universal cover $EG$ of $BG$ to $X$ under the $G$-action on $F(EG, X)$ given by $g$ in $G$ acts on a map $f$ in $F(EG, X)$ by sending it to $gfg^{-1}$. The $G$-equivariant map from $EG$ to a single point $*$ induces a natural map η: $X^G = F(*,X)^G$→$F(EG, X)^G$ from the fixed points to the homotopy fixed points of $G$ acting on $X$. Miller's theorem is that η is a weak equivalence for trivial $G$-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 $BZ/2$ as an unstable module over the Steenrod algebra.

Miller's theorem generalizes to a version of Sullivan's conjecture in which the action on $X$ 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 $G=Z/2$. 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 $(X^G)_p$ → $F(EG, (X)_p)^G$ is a weak equivalence when the order of $G$ is a power of a prime p, and where $(X)_p$ denotes the Bousfield-Kan p-completion of $X$. 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 $F(EG,X)^G$ before completion, and Lannes's proof involves his T-functor.

External links

• Book extract
• J. Lurie's course notes