My Math Forum Torus T^2 homeomorphic to S^1 x S^1

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March 19th, 2014, 05:25 PM   #1
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Torus T^2 homeomorphic to S^1 x S^1

[attachment=1:1wfzpmh8]Equation of Torus - Crossley - page 25.png[/attachment:1wfzpmh8]I am reading Martin Crossley's book, Essential Topology.

Example 5.43 on page 74 reads as follows:

[attachment=2:1wfzpmh8]Crossley - EXAMPLE 5.43 - page 74.png[/attachment:1wfzpmh8]

I am really struggling to get a good sense of why/how/wherefore Crossley came up with the maps f and g in EXAMPLE 5.43. How did he arrive at these maps?

Why/how does f map $S^1 \times S^1$ onto $T^2$ and how does one check/prove that this is in fact a valid mapping between these topological spaces.

Can anyone help in making the origins of these maps clear or perhaps just indicate the logic behind their design and construction? I am completely lacking a sense or intuition for this example at the moment ... ...

Definitions for $T^2$ and $S^1$ are as follows:

[attachment=1:1wfzpmh8]Equation of Torus - Crossley - page 25.png[/attachment:1wfzpmh8]

[attachment=0:1wfzpmh8]S^1 - Crossley - page 32.png[/attachment:1wfzpmh8]

My ideas on how Crossley came up with f and g are totally bankrupt ... but to validate f (that is to check that it actually maps a point of $S^1 \times S^1$ onto $T^2$ - leaving out for the moment the concerns of showing that f is a continuous bijection) ... ... I suppose one would take account of the fact that (x,y) and (x',y') are points of $S^1$ and so we have:

$x^2 + y^2= 1$ ... ... ... ... (1)

and

$x'^2 + y'^2 = 1$ ... ... ... ... (2)

Then, keeping this in mind check that

$((x' +2)x, (x' +2)y, y'$ is actually a point on the equation for $T^2$, namely:

$x^2 + y^2 + z^2 - 4 \sqrt{x^2 + y^2}= -3$ ... ... ... (3)

So in (3) we must:

- replace x by (x' +2)x
- replace y by (x' +2)y
- replace z by y'

and then simplify and if necessary use (1) (2) to finally get -3.

Is that correct? Or am I just totally confused ?

Peter
Attached Images
 Crossley - EXAMPLE 5.43 - page 74.png (53.8 KB, 35 views) Equation of Torus - Crossley - page 25.png (24.6 KB, 35 views) S^1 - Crossley - page 32.png (82.9 KB, 35 views)

 Tags homeomorphic, torus

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