Talk:Loop theorem

Klaus Johannson also has a tower-free proof of the loop theorem, which predates Rubinstein and Lackenby, I believe. See "On the loop- and sphere-theorem" in Low-dimensional Topology, the proceedings of the 1992 Knoxville conference. best, Sam nead 06:05, 24 November 2006 (UTC)[reply]

I expect that the Johannson proof also uses a hierarchy and is essentially Waldhausen's proof, just like the other ones you mentioned, although of course, the implementations can be quite different and elegant. Right now, I'm going through the Waldhausen stuff. I'll take a look at the Johannson stuff if I get a chance. --C S (Talk) 06:26, 24 November 2006 (UTC)[reply]

Yes, Johannson uses the Haken heirarchy. That isn't my point. My point is that [AR] was published in 2004, [L] was written in 1999, and [J] was published in 1992. I freely admit that [J] is the only tower-free proof I am familiar with. This is the first time I've heard of [W] -- is it available on-line? Sam nead 18:01, 24 November 2006 (UTC)[reply]

Well, if you've taken a look and can say a few words about it, please do so in the article! I'm not going to add a reference to something I've not taken a look at. Waldhausen's paper on the world problem was in the Annals, available through JSTOR. --C S (Talk) 20:38, 24 November 2006 (UTC)[reply]

Just took a look at the review on MathSciNet of the Johannson paper. The reviewer says the loop theorem is proved "in full generality". It's difficult for me to believe that this method of proof could really work for say, the second version in the article (due to Stallings). So what does "full generality" mean here? --C S (Talk) 21:10, 24 November 2006 (UTC)[reply]

I'll look at Waldhausen's Annals paper. I should have done that already. "Full generality" certainly does not mean "full generality" -- there is no mention of groups, normal or otherwise, in Johannson's paper. However, he does conclude that there is a non-singular disk with boundary lying in a regular neighborhood of the boundary of the given singular disk. Perhaps that is it? best, Sam nead 03:29, 25 November 2006 (UTC)[reply]

Basepoint?

The statement of the theorem would be clearer if at least some reference were made to basepoints for the various fundamental groups.50.205.142.50 (talk) 15:46, 25 May 2020 (UTC)[reply]