Talk:Alexander's trick

This article was nominated for deletion on October 6, 2005. The result of the discussion was keep.

Sources[edit]

I based my edit on [1], p. 12, and [2], Lemma 5.1. However, neither seems valuable as a reference. -- Jitse Niesen (talk) 21:48, 8 October 2005 (UTC)[reply]

Any book I look at seems to have it, so I will list several of them. Anyway, the Alexander trick proper refers to the statement about isotopy. The extension result (to the whole ball) is rather trivial and is along the lines of a homework exercise for people who just learned what the definition of a quotient space is. The isotopy result is more tricky. My experience is that most (but not all) people (especially book authors) will refer to the more complicated one as the trick and not even refer to the obvious result as Alexander's trick. Occassionally in papers, people will refer to the obvious result as the trick, but this is getting less and less common nowadays. So the article should be revised to state the more substantive result first and then as an aside mention that some people call the simpler result the same name.

Thurston's book, p. 124 Problem 3.2.10: (a) Show that a homeomorphism of the unit ball in R^n that is the identity on its boundary is isotopic to the identity. (Hint: comb all the tangles to a single point. This is called the Alexander trick.)

Rourke and Sanderson, p. 37 Proposition 3.22 (i): Let B^n, C^n be balls and h_0, h_1: B^n \rightarrow C^n homeomorphisms which agree on boundary of B^n, then h_0, h_1 are ambient isotopic mod boundary B^n.

Proof: (i) (Alexander trick) [....]

The result is also in other books such as Rolfsen and Bing's books on knot theory and 3-manifolds resp, but not given a name. Anyway, I gotta leave for half a week for a conference; this should be enough for people looking to help clean up the page. --C S 04:36, 14 October 2005 (UTC)[reply]