Let $M^3$ be a compact, connected and orientable smooth manifold with boundary $\partial M$. Suppose $\partial M$ consists of two connected components: $\partial_1 M$ and $\partial_2 M$.
The long exact sequence in homology (with $\mathbb{Z}$ coefficients) of the pair $(M, \partial M)$ is:
$$ 0 \to H_3(M, \partial M) \cong \mathbb{Z} \stackrel{f}{\to} H_2(\partial M) \cong \mathbb{Z}^2 \stackrel{g}{\to} H_2(M) \stackrel{h}{\to} H_2(M, \partial M) \stackrel{\partial}{\to} H_1(\partial M) \to \cdots$$
Can we identify the map $f$ and extract information about $g$, $h$ and $\partial$?
My thoughts: if $\partial M$ were connected, then $f : \mathbb{Z} \to \mathbb{Z}$ would be an isomorphism. In particular, $\partial : H_2(M, \partial M) \to H_1(\partial M)$ is injective if and only if $H_2(M) = 0$.