Homotopy Equivalences of 3-Manifolds

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 $1\le k\le A_M$ and $f^k$ is homotopic to a homeomorphism.