Let $M$ be a compact, orientable, irreducible 3-manifold with incompressible toroidal boundary (there might be more than one boundary component). Is it always possible to choose appropriate slopes on each boundary component, so that the resulting Dehn filling produces an irreducible closed 3-manifold?
3 Answers
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Yes. This follows from the characterisation of boundary slopes as a (union of) Lagrangian subspaces of the “symplectic space” of all slopes. See Theorem 1 of Hatcher’s paper.
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Corollary 1.2 of Gordon-Litherland's paper shows that for any toroidal boundary component, there are at most 6 Dehn fillings making the manifold reducible. Therefore, a generic filling of all boundary components is going to be irreducible.
As Sam pointed out, this improves an earlier result of Hatcher (which was already sufficient for the purpose of this question).
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$\begingroup$ Your link is to a paper by Gordon and Litherland, not Gordon and Luecke. Note also that the [GL] paper cites (an early version of) Hatcher's theorem on the first page. $\endgroup$– Sam NeadCommented Jul 7 at 11:32
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If I haven't misunderstood, this will be a direct corollary of Theorem 0.1 in Y.-Q. Wu: The reducibility of surgered 3-manifolds, Topology Appl. 43 (1992), 213-218.
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1$\begingroup$ This will deal with the case of one boundary component. Note that the machinery required to get a distance bound (as in the paper you cite) is much more demanding that the machinery required to get a "dimension bound" (as in my answer). $\endgroup$– Sam NeadCommented Jul 7 at 11:16