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

From: Lee Rudolph (lrudolph_at_panix.com)
Date: 08/28/04 

Date: 28 Aug 2004 12:46:09 -0400

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. On the one hand, D is homeomorphic
to the round 4-disk D^4 (you can take D to be the "unit" bidisk in R^4,
and then easily construct the homeomorphism explicitly), so its
boundary is homeomorphic to the boundary of D^4, namely, S^3.
On the other hand, the given product structure on D gives you
an expression of the boundary of D as the union of two solid
tori intersecting along their common boundary: one solid torus
is the Cartesian product of a round circle with a round 2-disk,
and the other is the Cartesian product of a round 2-disk with
a round circle.

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

Lee Rudolph

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