Is there an open connected orientable 3-manifold $M$ with the following properties:


1. $M$ admits a complete hyperbolic metric with finite hyperbolic volume.
2. $H\_{i}(M,\mathbb{Z})=0$ for any $i>0$.



Special kind of 3-manifolds
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](https://pi.math.cornell.edu/%7Ehatcher/3M/3Mdownloads.html) for the statement and proof of "one-half lives, one-half dies".]


