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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.

A weakly contractible space is a nonempty path-connected topological space $X$ with trivial homotopy groups ${\pi }_n(X,x)$ for every $n\ge 1$. Equivalently, the map $X\to \{\ast \}$ is a weak homotopy equivalence. For a CW complex, the Whitehead theorem then implies that $X$ is contractible.^[ref]

Every contractible space is weakly contractible: a contraction makes every based sphere map nullhomotopic. The real line is a simple example, contracted by $H(x,t)=(1-t)x$. In contrast, the circle fails the test because its fundamental group is $\mathbb{Z}$. For arbitrary topological spaces, weak contractibility alone does not guarantee a continuous contraction; the CW complex hypothesis in the stated implication supplies substantive control over the space.