Re: S^3 as a union of two solid tori

From: Lee Rudolph (lrudolph_at_panix.com)
Date: 08/30/04 

Date: 29 Aug 2004 20:13:18 -0400

Chan-Ho Suh <suh@math.ucdavis.nospam.edu> writes:

>In article <CztYc.12716$D7.4300@news-server.bigpond.net.au>, Paul C.
>Leopardi <leopardi@bigpond.net.au> wrote:
>
>> arzille wrote:
>>
>> > The book says ... How can I use the hint of the book?
>>
>> OK, I'll bite. Which book? What page?
>>
>> I'd rather read the original than your paraphrase, because your paraphrase
>> does not seem to make any sense.
>>
>
>It does make sense. See my reply to Lee Rudolph's reply to arzille.

And actually it makes sense in several ways; not only those in your
reply to my reply to arzille, but also as an approach to the special
case in dimensions 1-and-1 of the general (and just as easily proved
--with a mindless calculation instead of a helpful picture, even!)
result that the join of an m-sphere and an n-sphere is an
(m+n+1)-sphere. (For those playing along at home without their
score-cards, the "join" of topological spaces A and B is, roughly
but correctly, the union of a family of intervals parametrized
by AxB, where the interval I_{a,b} has endpoints a in A and b in
B, and is otherwise disjoint from A, B, and all the other intervals,
and where the union of all these intervals is given the only reasonable
topology. For instance, the join of an interval I and an interval
J is a solid tetrahedron; the locus of midpoints of the join intervals
is a square cross-section of the tetrahedron. Starting from there
and some simple facts about joins and circles, you can fairly quickly
show that the join of two circles C and D is a 3-sphere, and that
the locus of midpoints of the join intervals is a torus which bounds
a solid torus on each side.)

Lee Rudolph