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 onto 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 and 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 onto - 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:
... ... ... ... (1)
... ... ... ... (2)
Then, keeping this in mind check that
is actually a point on the equation for , namely:
... ... ... (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 http:///images/smilies/smile.png ?
Can someone please help?
NOTE: The above has also been posted on MHB
Re: Torus T^2 homeomorphic to S^1 x S^1
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.