Homology sphere

Remarks

The condition $H_0(X; \Z) \cong \Z$ says that $X$ is connected, and $H_n(X; \Z) \cong \Z$ says that $X$ is orientable.

Example: the Poincaré homology sphere $P = S^3 / I^\ast$, where $I^\ast \subset S^3$ is the binary icosahedral group of order $120$ acting by left multiplication. Its fundamental group $I^\ast$ is perfect, so $H_1(P; \Z)$, the abelianisation of $I^\ast$, is $0$; Poincaré duality and the universal coefficient theorem then give $H_2(P; \Z) = 0$ and $H_3(P; \Z) \cong \Z$.

Non-example: the lens space $L(p, q)$ with $p \ge 2$ is a closed orientable $3$-manifold, but $H_1(L(p, q); \Z) \cong \Z/p \neq 0$.

A simply connected homology $n$-sphere is homotopy equivalent to $S^n$, by the Hurewicz and Whitehead theorems.