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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. |
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8
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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. |
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4
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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". |
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