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$\begingroup$

How would you show that $\pi_n, n>1$ of the Klein bottle is the trivial group?

I was thinking Seifert-Van Kampen could be applicable?

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    Seifert-van Kampen doesn't say anything about the higher homotopy groups.2015-11-29

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The universal cover of the Klein bottle is $\mathbb{R}^2$. You can use lifting criteria from covering space theory to show that the Klein bottle and its covering spaces have the same higher homotopy groups.

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    @gabriel That's right.2012-11-15