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 ${\displaystyle G}$. The most elementary formulation, however, is in terms of the classifying space ${\displaystyle BG}$ of such a group. Roughly speaking, it is difficult to map such a space ${\displaystyle BG}$ continuously into a finite CW complex ${\displaystyle X}$ in a non-trivial manner. 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 ${\displaystyle BG}$ to ${\displaystyle X}$ is weakly contractible.

This is equivalent to the statement that the map ${\displaystyle X}$ → ${\displaystyle F(BG,X)}$ from X to the function space of maps ${\displaystyle BG}$ → ${\displaystyle X}$, not necessarily preserving the base point, given by sending a point ${\displaystyle x}$ of ${\displaystyle X}$ to the constant map whose image is ${\displaystyle x}$ is a weak equivalence. The mapping space ${\displaystyle F(BG,X)}$ is an example of a homotopy fixed point set. Specifically, ${\displaystyle F(BG,X)}$ is the homotopy fixed point set of the group ${\displaystyle G}$ acting by the trivial action on ${\displaystyle X}$. In general, for a group ${\displaystyle G}$ acting on a space ${\displaystyle X}$, the homotopy fixed points are the fixed points ${\displaystyle F(EG,X)^{G}}$ of the mapping space ${\displaystyle F(EG,X)}$ of maps from the universal cover ${\displaystyle EG}$ of ${\displaystyle BG}$ to ${\displaystyle X}$ under the ${\displaystyle G}$-action on ${\displaystyle F(EG,X)}$ given by ${\displaystyle g}$ in ${\displaystyle G}$ acts on a map ${\displaystyle f}$ in ${\displaystyle F(EG,X)}$ by sending it to ${\displaystyle gfg^{-1}}$. The ${\displaystyle G}$-equivariant map from ${\displaystyle EG}$ to a single point ${\displaystyle *}$ induces a natural map η: ${\displaystyle X^{G}=F(*,X)^{G}}$→${\displaystyle F(EG,X)^{G}}$ from the fixed points to the homotopy fixed points of ${\displaystyle G}$ acting on ${\displaystyle X}$. Miller's theorem is that η is a weak equivalence for trivial ${\displaystyle G}$-actions on finite-dimensional CW complexes. An important ingredient and motivation for his proof is a result of Gunnar Carlsson on the homology of ${\displaystyle BZ/2}$ 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 ${\displaystyle X}$ is allowed to be non-trivial. 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 ${\displaystyle G=Z/2}$. This conjecture was incorrect as stated, but a correct version was given by Miller, and proven independently by Dwyer-Miller-Neisendorfer,^[4] Carlsson,^[5] and Jean Lannes,^[6] showing that the natural map ${\displaystyle (X^{G})_{p}}$ → ${\displaystyle F(EG,(X)_{p})^{G}}$ is a weak equivalence when the order of ${\displaystyle G}$ is a power of a prime p, and where ${\displaystyle (X)_{p}}$ denotes the Bousfield-Kan p-completion of ${\displaystyle X}$. 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 ${\displaystyle F(EG,X)^{G}}$ before completion, and Lannes's proof involves his T-functor.^[7]

References

External links

• Gottlieb, Daniel H. (2001) [1994], "Sullivan conjecture", Encyclopedia of Mathematics, EMS Press
• Book extract
• J. Lurie's course notes