Noncompact homology spheres? Are all homology spheres compact? Are all generalized homology spheres compact? By a homology sphere I mean an $n$-manifold $X$ with same homology as the $n$-sphere. By a generalized homology sphere I mean the same with the assumption "$n$-manifold" replaced by "homology $n$-manifold".
If that helps, assume further that the spaces under consideration are simply connected.
 A: If $M$ is a connected, non-compact $n$-manifold, then $H_i(M;R)=0$ for $i\geq n$. For a proof, see Proposition 3.29 in Hatcher's Algebraic Topology book.
So, if you are going to have $H_n(M;R)=R$, $M$ had better be compact.
EDIT (to answer about homology $n$-manifolds):
A homology $n$-manifold is a finite dimensional, locally contractible space $X$ whose local homology groups $H_*(X, X-\{x\})$ are the local homology groups for $\mathbb{R}^n$ for every $x\in X$. In particular, $\mathbb{R}^n$ is a non-compact homology $n$-manifold.
A: The top dimensional cohomology of a connected manifold is non-zero if and only if the manifold is compact, so the answer is "yes" for homology spheres.
EDIT: As Mariano remarks below, "compact" should read "compact and orientable".
A: By a paper of Martio and Ryazanov, every connected, compact and simply connected homology $n$-sphere is homeomorphic to $S^n$, if $ \ n \geqslant 4$. By the affirmative answer to the Poincaré conjecture, this follows on dimension $3$ as well.
