Sullivan conjecture

Remarks

This form was proved by Haynes Miller in 1984, and is often called Miller's theorem. The proof uses the category of unstable modules over the Steenrod algebra: one shows that the relevant Ext groups vanish and uses the unstable Adams spectral sequence. The hypothesis that $X$ is finite-dimensional cannot be dropped: if $G \ne 1$, the identity map of $BG$ is not null-homotopic, since $\pi_1(BG) \cong G$ is nontrivial.

At work: every map from $B\Z/p$ to a finite-dimensional complex, such as a sphere $S^n$, is null-homotopic. Sullivan's conjecture also has a fixed-point form, proved independently by Carlsson, by Lannes, and by Dwyer, Miller and Neisendorfer. It says that, for a finite $p$-group acting on a finite complex, the fixed points agree with the homotopy fixed points after $p$-completion, under suitable hypotheses.

Going deeper

For a student

Miller's theorem was the starting point of the theory of Lannes' $T$ functor and unstable modules over the Steenrod algebra. These methods give a homotopy-theoretic description of maps out of $BG$ in many cases. Its fixed-point form compares the fixed points $X^G$ of a group action with the homotopy fixed points $X^{hG}$, and is parallel to the Segal conjecture for spectra.