Talk:Whitehead manifold

2-sphere or 3-sphere

Isn't the three dimensional sphere called S^2 not S^3 ? --anon

Well, S^2 is the 2-D sphere embedded in R^3, that is, the usual sphere. S^3 is the sphere in R^4, not in R^3. So everything is correct. Oleg Alexandrov 15:34, 8 August 2005 (UTC)[reply]

Why null-homotopic?

I don't understand this part: “Note that T₂ is null-homotopic in T₁, in particular avoiding the meridian of T₁”. I understand why T₂ is null-homotopic in T₁ but not at all how we could avoid the meridian. Looking at the picture of the Whitehead link, it looks to me that the two pieces are precisely knotted so as to make each one not null-homotpoic avoiding the other. (I know I'm wrong, here, but what I ask for is an explanation—in the article, that is.) --Gro-Tsen 09:16, 16 January 2006 (UTC)[reply]

I'm guessing you are picturing an isotopy, which is different. Remember that in a homotopy of the curve, for each time parameter the map does not need to be one to one, whereas for an isotopy it does. So once you allow the loop to pass through itself, I think you should be able to see an obvious homotopy that shrinks it to a point. I avoided giving more details as at the time, there was no picture of Whitehead link and so there was no good way of describing the null-homotopy; I merely stated the necessary property and left it as an exercise, so to speak.

I should note that the pic at Whitehead link is not really appropriate for this construction. There's another standard diagram where one component is the meridian of a solid torus and the other one (called the "Whitehead double") is inside the solid torus; that one would be better for the Whitehead construction. The process of replacing a solid torus by the Whitehead double is called Whitehead doubling. This is probably worth a short article on its own, since it's a nice operation that comes in handy in other contexts. There's also another doubling procedure called "Bing doubling". --C S (Talk) 09:53, 16 January 2006 (UTC)[reply]

Yes, now I see it: you're right, I was thinking in terms of isotopy rather than homotopy. Another way to make things clear is to point out that the fundamental group of the torus minus its meridian is Z² and the loop is chosen to have zero winding. Still, I'd appreciate it if you could add a few words of explanation (I prefer not to do it myself, being so very ignorant of homotopy and knots). And an article about Whitehead doubling would be most welcome. --Gro-Tsen 14:56, 16 January 2006 (UTC)[reply]

Sure, I will make the necessary edit; I like your comment about the winding number so I'll use that. But I don't understand your comment about the fundamental group. That picture and Whitehead double will have to wait though. --C S (Talk) 22:21, 16 January 2006 (UTC)[reply]

Our previous observation

"It follows from our previous observation, the Hurewicz theorem, and Whitehead's theorem on homotopy equivalence, that X is contractible."

I'm not clear what previous observation this refers to. I think this proof would be a bit clearer (to me, at least) if it were fleshed out what the nature of this observation is and what the maps are in the application(s) of Whitehead's theorem. vivacissamamente (talk) 22:07, 28 May 2013 (UTC)[reply]