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I'm trying to see the fundamental group of the Klein bottle minus a point without success. I know how to solve the torus minus a point giving a deformation retraction to the wedge sum of two circles.

My solution of the torus minus a point:

enter image description here

I need help here.

thanks a lot

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How would you solve the torus problem, and can you copy that solution to the Klein bottle case? –  Sanchez Jan 26 '13 at 3:48
    
@Sanchez yes of course, I'm doing this right now –  user42912 Jan 26 '13 at 3:53
    
@Sanchez is it ok? :) –  user42912 Jan 26 '13 at 4:03
    
it looks fine - so what problem do you meet when you translate that solution to this case? –  Sanchez Jan 26 '13 at 4:05
    
@Sanchez intuitively it seems a wedge sum of two circles as well, but it isn't. –  user42912 Jan 26 '13 at 4:08
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1 Answer

up vote 13 down vote accepted

We have two representations of the Klein bottle as a fundamental polygon. The first is: enter image description here

And the second is formed by cutting this polygon across the diagonal, flipping one piece and reattaching it to give essentially two real projective planes glued together:

enter image description here

You should be able to see that as CW complexes and a 2-cell attached according to the diagram, these both form the Klein bottle with non-orientable genus 2.

Removing a point from this 2-cell produces a space that deformation rectacts onto the 1-skeleton, which in both cases obviously forms the wedge sum of two circles and the fundamental group is $\Bbb{Z} * \Bbb{Z}$.

Let's see if we can develop some sort of physical intuition for this. If the point (or by deformation, hole) we remove is in the right place, we can embed this in $\Bbb{R}^3$ to get a physical intuition. enter image description here

Which then forms enter image description here

And you can see rather easily that this deforms to: enter image description here

Which obviously has the fundamental group of $\Bbb{Z} * \Bbb{Z}$, as this deformation retracts onto $S^1 \vee S^1$.

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Nice, although one of your bands in the last picture should have a half-twist. –  Grumpy Parsnip Feb 3 '13 at 21:52
    
It should. Although it doesn't really make a difference as far as $\pi_1$ is concerned. I'll fix it anyways. –  Sam DeHority Feb 3 '13 at 21:53
    
All fixed! Thanks Jim. –  Sam DeHority Feb 3 '13 at 22:00
    
thank you very much for your answer! –  user42912 Feb 5 '13 at 10:22
    
@user42912 You're most welcome, glad I could help. –  Sam DeHority Feb 6 '13 at 13:34
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