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The other day I found an article which proved the Jordan Curve Theorem using the van Kampen Theorem for the fundamental groupoid of a space. I was wondering if this technique could be extended to prove the more general Jordan-Brouwer Separation Theorem, using higher homotopy groupoids, the "van Kampen Theorem" for higher homotopy groupoids, or something else along these lines. I think it would be interesting if such an approach were possible.
Here's the article I mentioned: http://www.maths.ed.ac.uk/~aar/jordan/pb-jord.pdf
Here's the article I mentioned: http://www.maths.ed.ac.uk/~aar/jordan/pb-jord.pdf