Fundamental group determines homology groups for compact connected orientable 3-manifold

Statement
For a compact connected orientable manifold of dimension 3, knowledge of the isomorphism class of the fact about::fundamental group is sufficient to determine the isomorphism classes of all the homology groups.

Facts used

 * 1) uses::Hurewicz theorem: Used to get $$H_1$$ from $$\pi_1$$ as the abelianization.
 * 2) uses::Dual universal coefficients theorem: Used to get $$H^1$$ from $$H_1$$ as the torsion-free part.
 * 3) uses::Poincare duality theorem: Used to get $$H_2$$ from $$H_1$$. Also, confirms that $$H_0$$ and $$H_3$$ are isomorphic to $$\mathbb{Z}$$.