If two topological spaces have the same homotopy type, how do I prove that their complements in an ambient space are homeomorphic?
If two topological spaces have the same homotopy type, how do I prove that their complements in an ambient space are homeomorphic?
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1Well my specific example was that I have a 2-complex which is homotopy equivalent to the 2 sphere. This directly implies that the its complement in the 3-sphere is the union of two open 3-balls. How do I show that this is true? – 2011-02-22
3 Answers
I don't know how your question can be fixed so as to get a true statement, but as asked this is simply wrong. Take the $x$-axis in $\mathbb{R}^{2}$. It is contractible and its complement has two connected components. However, the complement of a point only has one connected component, so this is an easy example of two subspaces of the same homotopy type whose complements aren't homeorphic or even homotopic.
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1346rte: I can't see that page on Google Books, and my copy must be at the office. I suggest you edit your question to specifically explain the situation you're interested in. – 2011-02-23
Indeed, the whole subject of knot theory is interesting because different subsets of $R^3$ that are homeomorphic to each other (namely different knots, each homeomorphic to a circle) have non-homeomorphic complements in $R^3$. And knots are homeomorphic, not just homotopy equivalent.
Alexander duality is the closest to what you seem to want: http://en.wikipedia.org/wiki/Alexander_duality.