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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 $

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You get:

  • the solid torus,
  • all the Seifert manifolds 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.

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    • $\begingroup$ Hi Dr. Martelli, Thank you for your complete answer. I have never thought about Seifert fibered spaces before. Suppose I am in the latter case where both meridians are not glued along the fibres. If we take a basis of homology to be given by the meridian and the fiber of each torus, this allows us to give a matrix representation which records the maps $f_i$ on $H_1$. What property of the matrix determines whether a given fiber is exceptional or not? $\endgroup$ – user36931 Jul 18 '14 at 6:41
    • $\begingroup$ Is the correct way to calculate the multiplicity of the fiber to take $f_i^{-1}$ and then consider the intersection number of the image of the fiber of $P \times S^1$ with the meridian circle of $S^1 \times D^2$? $\endgroup$ – user36931 Jul 18 '14 at 9:14
    • $\begingroup$ The gluing of a solid torus is parametrized simply by looking at the image (p,q) of the meridian: the image of the longitude is useless here, see en.wikipedia.org/wiki/Dehn_surgery You get: (0,1) is fiber-parallel, and (1,q) is non-exceptional. $\endgroup$ – Bruno Martelli Jul 18 '14 at 10:04
    • $\begingroup$ Yes (answer to second question) the number "p" in (p,q) is precisely that geometric intersection $\endgroup$ – Bruno Martelli Jul 18 '14 at 10:05
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    $P\times S^1$ has an obvious fibration by circles and as long as your Dehn filling does not send a meridian to the fiber of that fibration, the Dehn filled manifold will again be a Seifert fibration. You may then use the classification of Seifert fibrations to give a name to your manifold.

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