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Danny Ruberman
Danny Ruberman
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Yes, there are conditions; basically you want to maintain the hypothesis of not having incompressible tori and annuli at each stage. A really good reference for this is Morgan's essay, "On Thurston's uniformization theorem for three-dimensional manifolds", in Morgan, John W.; Bass, Hyman, The Smith conjecture (New York, 1979), Pure Appl. Math. 112, Boston, MA: Academic Press, pp. 37–125. I'd suggest looking there.

(I edited out the "tori"; this will be automatic if the original manifold is atoroidal as it should be.)

Yes, there are conditions; basically you want to maintain the hypothesis of not having incompressible tori and annuli at each stage. A really good reference for this is Morgan's essay, "On Thurston's uniformization theorem for three-dimensional manifolds", in Morgan, John W.; Bass, Hyman, The Smith conjecture (New York, 1979), Pure Appl. Math. 112, Boston, MA: Academic Press, pp. 37–125. I'd suggest looking there.

Yes, there are conditions; basically you want to maintain the hypothesis of not having incompressible annuli at each stage. A really good reference for this is Morgan's essay, "On Thurston's uniformization theorem for three-dimensional manifolds", in Morgan, John W.; Bass, Hyman, The Smith conjecture (New York, 1979), Pure Appl. Math. 112, Boston, MA: Academic Press, pp. 37–125. I'd suggest looking there.

(I edited out the "tori"; this will be automatic if the original manifold is atoroidal as it should be.)

Source Link
Danny Ruberman
Danny Ruberman
  • 19.4k
  • 1
  • 60
  • 97

Yes, there are conditions; basically you want to maintain the hypothesis of not having incompressible tori and annuli at each stage. A really good reference for this is Morgan's essay, "On Thurston's uniformization theorem for three-dimensional manifolds", in Morgan, John W.; Bass, Hyman, The Smith conjecture (New York, 1979), Pure Appl. Math. 112, Boston, MA: Academic Press, pp. 37–125. I'd suggest looking there.