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.
Best Answer
Sounds like you have the prerequisites, though it might help to learn some more homology. Most of the older arguments rely on some piece-wise linear topology, but you can pick that up along the way (or think of things smoothly).
When I learned the Loop Theorem in grad school, I used a combination of Calegari's notes, Papakyriakopoulos' original paper, Hatcher's notes on 3-manifolds, and Stallings' paper and text. Eventually I found it in Hemple's text, but this was well after I learned the proofs. Have fun with it! I've always loved these theorems, even if it took a bit of digging to find all the ins and outs of their proofs.