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Does anyone know what this group is. Just want to know what the group is. Thanks.

Brian M. Scott
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doncarlos
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  • think of the case, when Klein bottle has one point removed. You can enlarge the hole all along to get a regular strip loop and a Mobius strip loop glued at one point. – Yimin Feb 20 '13 at 03:56

1 Answers1

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It is a free group generated by $3$ letters. If you prefer a proof. See here for one point off case.

fundamental group of the Klein bottle minus a point

And if you continue to minus points, you will see, whenever you removed a point, you generated a new circle attached to it.

Community
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Yimin
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  • I calculated the fundamental group of the torus with two points removed. I know the group is $F_3$. However, the second calculation I got the group. $ <a, b, c, d|aba^{-1}b^{-1}cdc^{-1}d^{-1}>$ Is this equal to $F_3?$ And if so why? – doncarlos Feb 20 '13 at 04:16
  • are you sure the fundamental group is $\langle a,b,c,d | aba^{-1}b^{-1}cdc^{-1}d^{-1}\rangle$? I cannot see it is a free group either. – Yimin Feb 20 '13 at 15:58