The Sullivan conjecture
Haynes Miller · Bulletin of the American Mathematical Society · 1983
THEOREM 0. Let G be a locally finite group with classifying space BG and let X be a connected finite dimensional CW complex.Then the space of pointed maps from BG to X is weakly contractible: Wi (map+(BG,X)) = 0 for all i > 0. In particular, every map from BG to X is null-homotopic.