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 $



What 3-manifolds can be obtained by gluing $ S^1 \times P $ and two copies of $S^1 \times D^2$
You get:



- the solid torus,
- all the [Seifert manifolds](http://en.wikipedia.org/wiki/Seifert_fiber_space) fibering over the disc with two exceptional fibers; 
- the connected sum $(D\times S^1) \# L(p,q)$ of a solid torus and a lens space $L(p,q)$, for all coprime $(p,q)$.



The proof goes as follows. Consider the filling meridians of the two solid tori you attach. If one meridian is attached along the fiber of $P\times S^1$, then the resulting manifold is the connected sum $(D\times S^1) \# (D \times S^1)$ of two solid tori: this is a nice exercise in 3-dimensional topology. The attaching of the second solid torus then produces $(D^2 \times S^1) \# L(p,q)$.


If both meridians are not glued along the fibers, then the fibration of $P\times S^1$ extends to a Seifert fibration over the disc with at most two exceptional fibers. If at least one fiber is not exceptional, then you actually get a solid torus.


