= Sullivan conjecture
{c}
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>:
$$
\pi_n\operatorname{Map}_*(BG,X)=0\quad(n\ge1),\qquad \pi_0\operatorname{Map}_*(BG,X)=\{*\}.
$$
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.<The Sullivan conjecture on maps from classifying spaces by Haynes Miller>[Theorem A, printed page 39]{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.
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