# Compact, orientable $3$-manifolds with trivial $H_2$ and nontrivial relative $H_2$ [closed]

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.