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Does this mean that the first homotopy group in some sense contains more information than the higher homotopy groups? Is there another generalization of the fundamental group that can give rise to non-commutative groups in such a way that these groups contain more information than the higher homotopy groups?

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    Interesting question. I don't think that there's necessarily less information in an abelian group: the commutativity of the higher groups is really a consequence of a geometrical fact about spheres, and not really a restriction. Might also be worth considering the relative homotopy groups: $\pi_1$ becomes a set and $\pi_2$ non-abelian. But these are just thoughts, not a real answer.2011-03-04

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