- #1
PsychonautQQ
- 784
- 10
edit: fixed typo's andrewkirk pointed out, oops
I can cover the projective plane with 2 open sets U,V where each of these neighborhood contains the point that is missing, and the intersection of these two neighborhoods will be simply connected.
I was then hoping to invoke the Seifert-Van-Kampen theorem to see that the fundamental group would be isomorphic to the Free product of two infinite cyclic groups, because the fundmental group of U and V would both be isomorphic to the fundmanetal group of ##S^1##. This is what I thought anyway.
I am a bit concerned because this is exactly the result I arrived at for a closed disk with 2 punctured points, and it seems it would be the same thing for a sphere with 2 punctured points. Am I doing something wrong here? Thanks PF!
I can cover the projective plane with 2 open sets U,V where each of these neighborhood contains the point that is missing, and the intersection of these two neighborhoods will be simply connected.
I was then hoping to invoke the Seifert-Van-Kampen theorem to see that the fundamental group would be isomorphic to the Free product of two infinite cyclic groups, because the fundmental group of U and V would both be isomorphic to the fundmanetal group of ##S^1##. This is what I thought anyway.
I am a bit concerned because this is exactly the result I arrived at for a closed disk with 2 punctured points, and it seems it would be the same thing for a sphere with 2 punctured points. Am I doing something wrong here? Thanks PF!
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