Proof of Jordan-Brouwer Separation Theorem Using Homotopy Theory?

[jgens]

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

 

[pat_connell]

Unfortunately, it does not seem that the approach in this article can be extended to prove the Jordan-Brouwer Separation Theorem. The van Kampen Theorem only applies to the fundamental groupoid of a space, which is not powerful enough to capture the essential features of the Jordan-Brouwer Separation Theorem. The Jordan-Brouwer Separation Theorem requires a more sophisticated tool than the van Kampen Theorem, namely the theory of homology and cohomology.