# Talk:Lens space

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Field: Topology

## Amazing

I'm amazed. Although this article contains many words and quite a lot of math, it does not seem to say anything at all. How is anyone to understand what a lens space is from this article? ᓛᖁ 20:15, 3 May 2005 (UTC)

## article needs correct condition for isomorphism of L(p,q)'s

I removed what I believe is the false statment that L(p,q) is diffeomorphic to L(p',q') iff p=p' and q=q'. I believe the correct condition is that p=p' and either q=q' or q=1/q' (where the arithmetic and equalities are mod p). Actually, q = -q' also yields the same lens space if one deals with unoriented 3-folds, as presumably is the default assumption.

If someone can confirm this information, they should add it to the main article.

- Rafael Sorkin

You're absolutely correct! What's funny is that the intro paragraph makes clear a very old counterexample by Alexander. Anyway, the misinformation seems to have been added after I had stopped watching the article. But I will add the correct statement now. --C S (Talk) 13:42, 21 February 2006 (UTC)

## More Definitions

I added some more definitions that I have come across (and in fact can be located in "A Short Survey of Lens Spaces" by Watkins)LkNsngth (talk) 02:04, 8 April 2008 (UTC)

Bravo, great work, 64.48.78.41 05:08, 27 July 2007 (UTC)Steve Pax

I think care needs to be taken when looking at the article by Watkins, I'm basing my undergraduate thesis on it, and I've found one or two errors in his workings so far. - Goldcont —Preceding unsigned comment added by 93.96.214.175 (talk) 14:51, 28 February 2010 (UTC)

## Definition

I found the original definition painful to read and somewhat imprecise. Hence I completely rewrote it. Instead of giving just the definition for arbitrary dimensions, I first define the 3-dimensional lens spaces as this is what most readers will be interested in (including me). The general case is then just a simple modification. I hope that it is now is easier to read and understand.

The infinite dimensional case needs a bit more care. The current version is somewhat laconic. Moreover, I am not sure whether a wikipedia-article needs this generality. Thus I have deleted this case in my edit:

<We can also define the infinite-dimensional lens spaces as follows. These are the spaces L(p;q_1,q_2,\ldots) formed from the union of the increasing sequence of spaces L(p;q_1,\ldots,q_n) for n=1,2,\ldots. As before, the q_1,q_2,\ldots must be coprime to p.>

I am also not sure whether the additional section on the 3-dimensional case is now really needed. If noone objects, I will delete it in a few weeks.

Robert 138.246.7.176 (talk) 15:20, 8 June 2010 (UTC)