Following Lemma 2.7 from Vogtman's Rational Homology of Bianchi Groups, I want to define cuspidal cohomology as $$H_{\mathrm{cusp}}(M)=\frac{H_1(M)}{i_*(H_1(\partial M))}$$ where $i:\partial M\to M$ is the inclusion and I take integral coefficients. The motivation is that Abelian covering spaces such that the preimage of a cusp consists of disjoint copies of the same cusp are corresponding to exactly those holonomies $\pi_1(M)\to G$ that factor through the cuspidal cohomology.

I verified for all hyperbolic cusped 3-manifolds in the SnapPea census that if $H_1(M)=\mathbb{Z}^n\oplus T$ where $T$ is torsion, then $H_\mathrm{cusp}(M)=\mathbb{Z}^{n-\mbox{number of cusps}}\oplus T$. This is to be expected rationally by half-lives-half-dies. It seems to also hold integrally for hyperbolic 3-manifolds.

Is there an argument that this holds for all hyperbolic cusped 3-manifolds? (And maybe a counterexample if you take a non-hyperbolic manifold)

Following Lemma 2.7 from Vogtmann's Rational Homology of Bianchi Groups, I want to define cuspidal cohomology as $$H_{\mathrm{cusp}}(M)=\frac{H_1(M)}{i_*(H_1(\partial M))}$$ where $i:\partial M\to M$ is the inclusion and I take integral coefficients. The motivation is that Abelian covering spaces such that the preimage of a cusp consists of disjoint copies of the same cusp are corresponding to exactly those holonomies $\pi_1(M)\to G$ that factor through the cuspidal cohomology.

I verified for all hyperbolic cusped 3-manifolds in the SnapPea census that if $H_1(M)=\mathbb{Z}^n\oplus T$ where $T$ is torsion, then $H_\mathrm{cusp}(M)=\mathbb{Z}^{n-\mbox{number of cusps}}\oplus T$. This is to be expected rationally by half-lives-half-dies. It seems to also hold integrally for hyperbolic 3-manifolds.

Is there an argument that this holds for all hyperbolic cusped 3-manifolds? (And maybe a counterexample if you take a non-hyperbolic manifold)