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I hope I am not missing something obvious: can't we use the Jordan Curve Thm. to show

that the homology H_n(R^n) of R^n is trivial ? How about showing that Pi_n(R^n) is trivial?

It seems like the def. of cycles in a space X is geenralized by continuous , injective maps f: S^n -->X . When X=R^n, JCT says that f(S^n) separates R^n into 2 regions, which can be seen as saying that f(S^n) bounds, so that every cycle bounds, and then the homology is trivial.

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# Using Jordan Curve Thm to Show H_n(R^n) Trivial?

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