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

From: Chan-Ho Suh (suh_at_math.ucdavis.nospam.edu)
Date: 08/28/04 

Date: Sat, 28 Aug 2004 15:46:44 -0700

In article <cgqcsh$8u0$1@panix2.panix.com>, Lee Rudolph
<lrudolph@panix.com> wrote:

> arzille <arzille@ubc.edu> writes:
>
> >Dear all,
> > Can anyone explain why S3 can be expressed a a union of two solid
> >tori (S1 x 2-Disc) with an embedded torus (S1 x S1) as a common
> >boundary? The book says it follows easily by expressing R3- {solid
> >torus} as a unions of circle and a straight line and then add a point at
> >the infinity. How can I use the hint of the book?
>
> I think an easier way to see this is to consider the cartesian
> product D of two round 2-disks.

An even easier way is to consider S^3 as the union of two 3-balls.
Picture S^3 as R^3 plus infinity, and put one ball at the origin.
We're going to show this picture of S^3 as the union of these 2 balls
is the same as the union of two solid tori.

Take a solid torus and embed it in a standard way into R^3 plus
infinity. Now fill up the hole in the solid torus with a plug
(homeomorphic to D^2 x I). Voila, it's a 3-ball. The complement of
this 3-ball is clearly a 3-ball also. So the complement of our
original solid torus is a 3-ball with the plug glued on; each end of
the plug glues on the boundary of the 3-ball to give a solid torus (it
can't be a solid Klein bottle).

Thus this *particular* gluing of two solid tori gives S^3. If you pick
different homeomorphisms from the boundary of one solid torus to the
other as gluing maps, you'll in general get something different from
S^3. A lot of somethings. These are called lens spaces. This is
explained in Rolfsen's book.

The same approach shows that S^3 is the union of two genus g
handlebodies for any g. Just plug up each hole in a standardly
embedded genus g handlebody.

As for your hint:

You can get a picture of S^3 - solid torus as foliated by circles in
the way the hint describes by foliating the other torus.

A nice way to see if you get what's going on is to see if you can see a
sweep-out of tori sweeping between two circles, each of which is the
"core" of a solid torus.

[...]
>
> Since you're at UBC, why don't you go talk to Dale Rolfsen?
>

Maybe it's homework from Rolfsen :-)

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