Fundamental groups of the Moebius strip and the cylinder.
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Show that the Mobius strip and the cylinder both have fundamental group Z.
We can use the following theorem:
If G acts on X, pi1(X) = {e}, and for all x elements of X there exists Ux neighborhood of X such that Ux intersection g(Ux) = empty set for all g elements of G{e}, then pi1(XG) is homeomorphic to G.
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Solution Summary
We show that the fundamental groups of both the Moebius strip and the cylinder are Z.
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OK, take X=R^2 and define the action of Z on it as
(n, (x,y)) = (x+2pi n, y).
This action is properly discontinous. That is, for every (x,y) there's a neighborhood, U_e(x,y) such that the intersection U and (n,U) is not ...
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