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 Posted: Sat May 03, 2008 5:44 pm |
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I'm trying to work out the following variation of a problem from big
spivak, pg. 296. The problem in this form comes from a practice
qualifying exam from last year.
Let M^m and N^n are two compact oriented submanifolds of R^N, where N=n
+m+1, and M\cap N=\emptyset. Let \alpha be the map \alpha : MxN---->
S^{N-1} given by
\alpha(m,n)=(m-n)/|m-n|,
and define L(M,N)=deg\alpha.
(1) If M and N are contained in disjoint open balls of R^N, show that
L(M,N)=0.
(2) If {M_t: t\in[0,1]} and {N_t: t\in[0,1]} are differentiable
homotopies that are disjoint for all t, show that
L(M_0,N_0)=L(M_1,N_1).
I've been working on it about a day now, but can't seem to get
anywhere. Any help on this would be greatly appreciated.
Thanks in advance,
Petra |
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