Klein Bottle homeomorphism

Dear math experts,

I am trying to show that the Klein bottle (the square with opposite edges
identified, where two opposite arrows are in the same direction, and two
opposite arrows in the opposite direction) is homeomorphic to an annulus
whose antipodal points on the outer circle are identified, and whose
antipodal points on the inner circle are identified.

I have thought about it a bit, and I'd like to use the following strategy :

Let S be a square, A be an annulus, and let Klein bottle be S / ~ , where ~
was defined above, and let the second space above be A / ~ where ~ was
defined above as well.

I'd like to construct a commutative diagram where S ---> S / ~, A ----> A /
~ are canonical, and then I will define maps f : S ----> A and F : S /
~ -----> A / ~.

I decided that the most natural thing for f would be just exponentiation in
the complex plane. exp takes a square to an annulus (where the height and
length of the square is 2pi).

You can probably see the ensuing problem. If I identify opposite edges on
the square in the complex plane as I explained above (one pair using the
same arrows, the other pair using opposite arrows), I run into problems.
What happens after I exponentiate i s that I end up identifying an point on
the outer circle of the annulus, with the corresponding point on the inner
circle. This map also ends up identifying purely real points on the annulus
in a funny way.

I don't really want any hints, I guess I would just like to know if I am
close. I feel like my strategy here is very natural, and the map f :
S ----> A is also very natural. But I feel like my map f may not be correct
and I should probably find a different map.

Thank you for your thoughts,

James


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