Let P denote the pair of pants e.g. a sphere minus three small discs $D_1$,$D_2$,$D_3$ about marked points $x_1,x_2,x_3$. I then consider $P \times S^1$. We have boundary components $T_1$,$T_2$,$T_3$. Fix two homeomorphisms $f_1: \partial(S^1 \times D^2) \to T_1 $ and $f_2: \partial(S^1 \times D^2) \to T_2$.

What oriented three manifolds with boundary can be obtained by gluing $S^1 \times D^2$ along these maps (up to homeomorphism)? Namely what are the possible homeomorphism types of manifolds:

$S^1 \times D^2 \sqcup_{f_1} P \times S^1 \sqcup_{f_2} S^1 \times D^2 $