  Let $M$ be an oriented closed $3$-manifold. We prove that there exists a
constant $A_M$, depending only on the manifold $M$, such that for every
self-homotopy equivalence $f$ of $M$ there is an integer $k$ such that $1 \leq
k \leq A_M$ and $f^k$ is homotopic to a homeomorphism.
