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 ${\displaystyle G}$. 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. 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.

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