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jn
science forum beginner

Joined: 11 Jul 2006
Posts: 1

Posted: Tue Jul 11, 2006 1:52 pm    Post subject: Disconnecting the sphere

Can someone tell me if exist two subsets A and B of
S^m={x1^2+...+xn^2=1} such that:

A and B are homeomorphic
S^m\A is connected but S^m\B is disconnected
Lee Rudolph
science forum Guru

Joined: 28 Apr 2005
Posts: 566

Posted: Tue Jul 11, 2006 3:01 pm    Post subject: Re: Disconnecting the sphere

"jn" <jncunha17@yahoo.com> writes:

 Quote: Can someone tell me if exist two subsets A and B of S^m={x1^2+...+xn^2=1} such that: A and B are homeomorphic S^m\A is connected but S^m\B is disconnected

At least if A and B are minimally "nice", that can't happen,
by Alexander Duality (which probably should be called
"Alexander-Spanier Duality" unless A and B are very nice
indeed), which says that H_k(S^m\X;Z) is isomorphic to
H_{m-k-1}(X;Z), where H_* is reduced homology (say, Cech
homology to allow for less-nice sorts of things); in
your case, H_0(S^m\A;Z) is 0 and H_0(S^m\B;Z) isn't,
so H_{m-1}(A;Z) is 0 and H_{m-1}(B;Z), so A isn't
homeomorphic to B.

It's conceivable to me that all kinds of hell can break
loose if A and B aren't closed, or even if they are but
are really nasty locally.

Lee Rudolph
(it's also conceivable to me that I should have said
"cohomology" earlier, but certainly Alexander wouldn't
have done any such thing, so I won't either)

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