Let $X$ be a $\mathbb{Z}$-homology $n$-sphere, i.e., a closed manifold with $\mathbb{Z}$-homology groups of the standard $n$-dimensional sphere. If $X$ is simply-connected, it is not difficult to see (and not difficult to look up on the internet) that such a space is in fact homotopy equivalent (hence homeomorphic, by the Poincare conjecture) to the $n$-sphere.

Now assume that $X$ is a $\mathbb{Z}\_p$-homology sphere for some prime $p$. Is it true that if $X$ is simply-connected, then $X$ is homotopy equivalent to the $n$-sphere?

EDIT: OOPS, sorry! As pointed out in the comments, the first sentence didn't make any sense before. Added simply-connecteness as an assumption.