From: Chan-Ho Suh <csuh@math.ucla.edu>
Subject: Re: 3-sphere and donuts
Date: Fri, 30 Nov 2001 16:48:10 -0800
Newsgroups: sci.math
Summary: Viewing the 3-sphere is the union of two tori.

Nobuo Saito wrote:

> Robin Chapman <rjc@maths.ex.ac.uk> wrote:
> >Think about your average doughnut sitting in 3-space.
> >Ask yourself, what's its complement in (the 1-point
> >compactified) 3-space?
>
> Yes, someone wrote something like that.
> I tried to visualize it and vaguely I got a picture.
> But how do I convert it to a rigorous proof?

You can write explicit equations for it.  It helps to think of the
3-sphere as the points of length 1 in CxC, where C denotes the complex
numbers.

Alternatively (and my preferred method), you can think of S^3 = S^1 *
S^1, where * indicates "join".
==============================================================================

From: lrudolph@panix.com (Lee Rudolph)
Subject: Re: 3-sphere and donuts
Date: 30 Nov 2001 10:24:44 -0500
Newsgroups: sci.math

genkisaito@hotmail.com (Nobuo Saito) writes:

>Someone wrote the 3-sphere is the union of two donuts.
>I don't know the proof.
>Would anyone please post it or a clue?

What is the cartesian product of two 2-dimensional disks?
What is its "boundary" (in an obvious sense; for instance,
given that each of the original disks was in R^2, its
topological boundary in R^4)?  How does the product of
their two boundaries, a torus, sit inside it?

Lee Rudolph
==============================================================================

From: "Robin Chapman" <rjc@maths.ex.ac.uk>
Subject: Re: 3-sphere and donuts
Date: Sun, 2 Dec 2001 09:55:19 +0000 (UTC)
Newsgroups: sci.math

"Nobuo Saito" <genkisaito@hotmail.com> wrote in message
news:1zsx89ppebws@legacy...

> Robin Chapman <rjc@maths.ex.ac.uk> wrote:
> >Think about your average doughnut sitting in 3-space.
> >Ask yourself, what's its complement in (the 1-point
> >compactified) 3-space?
> 
> Yes, someone wrote something like that.
> I tried to visualize it and vaguely I got a picture.
> But how do I convert it to a rigorous proof?

Let S = {(a,b,c,d): a^2 + b^2 + c^2 + d^2 = 1}
be the 3-sphere. Let T_1 = {(a,b,c,d) in S: a^2 + b^2 <= 1/2}
and T_2 = {(a,b,c,d) in S: a^2 + b^2 >= 1/2}. Show that
T_1 and T_2 are solid tori (homeomorphic to D^2 x S^1) and
they meet in a torus.

Robin Chapman

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