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
"AlexanderSpanier Duality" unless A and B are very nice
indeed), which says that H_k(S^m\X;Z) is isomorphic to
H_{mk1}(X;Z), where H_* is reduced homology (say, Cech
homology to allow for lessnice 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_{m1}(A;Z) is 0 and H_{m1}(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) 
