Thread: Torus T^2 homeomorphic to S^1 x S^1

1. Torus T^2 homeomorphic to S^1 x S^1

I am reading Martin Crossley's book, Essential Topology.

Example 5.43 on page 74 reads as follows:

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 $\displaystyle S^1 \times S^1$ onto $\displaystyle T^2$ 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 $\displaystyle T^2$ and $\displaystyle S^1$ are as follows:

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 $\displaystyle S^1 \times S^1$ onto $\displaystyle T^2$ - 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 TEX] S^1 [/TEX] and so we have:

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

and

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

Then, keeping this in mind check that

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

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

NOTE: The above has also been posted on MHB

2. Re: Torus T^2 homeomorphic to S^1 x S^1

Bernhard,
When you post the same question on another site, I will not answer in detail. Besides, this expressly violates a rule of MHF. However here's a hint:

How do you get a torus? Answer, for one such torus, rotate the unit circle centered at (2,0) on the x-axis about the y-axis. Write down the parametric equations for the resulting surface and the homeomorphism should be clear.