Re: Maps of degree 1 from any n-dimensional compact manifold to S^n
From: Ron Sperber (ronsperber_at_optonline.net)
Date: 01/22/05
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Date: Fri, 21 Jan 2005 19:49:01 -0500
s wrote:
> I'm trying to prove that :
> Given a compact manifold of dimension n, there exists a map f:M-> S^n of
> degree 1.
> I'm told that you can prove this using local degree. However, I dont see
> how I can even construct a smooth map (which must be surjective o/w
> degree=0) from M to S^n.
>
Constructing a map is easy. Let U be a subset of M homeomorphic to an
open subset of R^n. Let D be a subset of U homeomorphic to D^n. Let
f:M->S^n be the map that maps Int(D) to S^n-{north pole}
homeomorphically and sends everything else to the north pole.
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