Is there an open connected orientable 3-manifold $M$ with the following properties:
- $M$ admits a complete hyperbolic metric with finite hyperbolic volume.
- $H_{i}(M,\mathbb{Z})=0$ for any $i>0$.
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6$\begingroup$ Such a manifold would be the interior of a compact orientable 3-manifold $N$. By half-lives, half-dies we must have $H_1(\partial N) = 0$, so each component of $\partial N$ is a 2-sphere. Since the hyperbolic metric has finite-volume, it can't have any 2-spheres as a boundary component, so $\partial N = \emptyset$. It follows that $M$ is closed, so $H_3(M) = \mathbb{Z}$. $\endgroup$– Andy PutmanCommented Aug 26, 2021 at 20:06
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1 Answer
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No. Suppose that $M$ is a finite volume oriented hyperbolic three-manifold. In the closed case, as $M$ is oriented, we have $H_3(M, \mathbb{Z}) \cong \mathbb{Z}$ generated by the fundamental class. In the open case, $M$ has torus cusps. Appealing to "one-half lives, one-half dies" we find that $M$ has non-trivial (in fact infinite) $H_1$.
[See Lemma 3.5 of Hatcher's notes on three-manifolds for the statement and proof of "one-half lives, one-half dies".]
