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I am looking for examples of compact, connected and orientable $3$-manifolds $M$ with boundary such that $H_2(M; \mathbb{Z}) = 0$ and $H_2(M,\partial M;\mathbb{Z}) \neq 0$. How big is this class? Is it finite, countable, uncountable?

I find this question interesting because if $M$ is a manifold having the above properties and $(\Sigma, \partial \Sigma) \subset (M, \partial M)$ is an embedded submanifold with boundary representing a nontrivial element of $H_2(M,\partial M;\mathbb{Z})$, then $\partial \Sigma$ represents a nontrivial element of $H_1(\partial M;\mathbb{Z})$. This follows from the fact that the map $H_2(M,\partial M;\mathbb{Z}) \to H_1(\partial M;\mathbb{Z})$ is injective is this case.

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