fundamental group of the Klein bottle minus a point

Solution 1:

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

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:

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.

Which then forms

And you can see rather easily that this deforms to:

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