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As part of my directed studies project, my advisor has suggested that I completely understand the proof of the Sphere Theorem and the Loop Theorem in 3-manifold theory and explain it to him. I have done an introductory and advanced course in general topology, and introductory courses in algebraic topology (homotopy theory, fundamental groups and simplicial homology theory). I've also spent the past month studying surgery on 3-manifolds through Knot Theory and the proof of the Lickorish Twist Theorem and the Lickorish Wallace Theorem from Dale Rolfsen's book Knots and Links.

My question is that what other prerequisites would I need to understand both of these theorems. Any other resources (lecture notes, video lectures etc.) that explain these two theorems in detail would also be appreciated. Rolfsen suggests John Stallings book and paper for these two theorems.

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  • $\begingroup$ Also posted on MSE math.stackexchange.com/q/4733233/116321 $\endgroup$
    – ZSMJ
    Commented Jul 19, 2023 at 10:23
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    $\begingroup$ Wouldn't it be easier to just read the proofs and see what (if anything) you are missing? Sounds like you already have sufficient background. $\endgroup$
    – Igor Belegradek
    Commented Jul 19, 2023 at 13:49
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    $\begingroup$ The usual method of reading research papers is to pick up necessary background as you go. This is normal. Don't let this discourage you. $\endgroup$
    – Igor Belegradek
    Commented Jul 19, 2023 at 18:56
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    $\begingroup$ Sounds like mostly you're missing material from a slightly more advanced algebraic topology course. You should be able to find that material nicely explained in Hatcher's Alegraic Topology or Munkres's Elements of Algebraic Topology. Of course you may need other things as well, but those will certainly cover Universal Coefficients and Poincare Duality. $\endgroup$
    – Greg Friedman
    Commented Jul 21, 2023 at 23:43
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    $\begingroup$ Andrew Casson has written lecture notes proving the sphere, disk, annulus, and torus theorems using "PL minimal surface theory" (following Jaco and Rubinstein). The notes can be found here: homepages.warwick.ac.uk/~masgar/casson.html $\endgroup$
    – Sam Nead
    Commented Sep 27, 2023 at 13:12

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On Marc Lackenby’s webpage you can find notes on 3-manifold topology (Michaelmas 1999). The proof of the loop theorem in Chapter 9 uses special hierarchies (instead of Papakyriakopoulos’ towers) following an approach of Johansson. The machinery of special hierarchies is maybe involved to set up, but it pays off to prove the loop theorem, topological rigidity of Haken 3-manifolds, and solving the word problem and homeomorphism problems.

There are many possible approaches to the loop and sphere theorems now available. Stallings gave a proof of the sphere theorem as a corollary of the loop theorem using his theory of ends of spaces and groups acting on trees. See also Jaco-Rubinstein. One may also deduce the loop theorem from the sphere theorem by doubling.

These theorems also follow as corollaries of the geometrization theorem. The proof of geometrization via Ricci flow makes use of minimal surface techniques that were developed originally by Meeks and Yau in the proof of the equivariant loop and sphere theorems (subsumed by the orbifold theorem).

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