Sullivan conjecture

A standard form of the Sullivan conjecture, proved by Haynes Miller, states that for a finite group $G$ and a connected finite-dimensional CW complex $X$, the pointed mapping space from the classifying space of a discrete group $BG$ to $X$ is a weakly contractible space:Thus every basepoint-preserving continuous map $BG\to X$ is null-homotopic, and higher homotopy families of such maps are trivial as well. Miller's Theorem A allows the more general assumption that every finitely generated subgroup of $G$ is finite.^[ref]

The finite-dimensional target hypothesis is substantial. For $G=\mathbb{Z}/2\mathbb{Z}$, maps from $BG$ to a finite-dimensional sphere satisfy the theorem, while the identity map of $BG$ is not null-homotopic: it induces the identity on its nontrivial fundamental group. That target $BG$ is infinite-dimensional. The theorem concerns these unstable mapping spaces; it cannot simply be transferred to every stable mapping construction.