A homology sphere is an $n$-dimensional topological manifold that has the same integral homology as the $n$-sphere.
Let $X$ be an $n$-dimensional topological manifold. Then $X$ is a homology sphere if
$$H_k(X, \mathbb{Z}) = \begin{cases} \mathbb{Z} & k = 0, n\\\ 0 & \text{otherwise}. \end{cases}$$
The most notable example of a homology sphere is the Poincaré homology sphere which has non-trivial fundamental group. If one were to replace the simply-connected hypothesis of the Poincaré conjecture with the vanishing of degree one homology, it would no longer be true (with the Poincaré homology sphere being a counterexample).
