I'm trying to understand a paper by Kawauchi and Matumoto, 'An estimate of infinite cyclic coverings and knot theory.' In one of their proofs they have a manifold $E$ which is the complement of a codimension-2 $n$-knot, with its infinite cyclic cover $\widetilde{E}$. They show that $\widetilde{H}_{*}(\widetilde{E};\mathbb{Z})=0$. Then they claim that because of this we must have that $E$ is homotopy equivalent to $S^{1}$.

It is clear that this shows $E$ is homologically equivalent to $S^1$, but why do we also get the result that it is homotopy equivalent?