Re: Homotopic maps

Your claim is correct at least for flat embeddings. Here's why:

Let U be an open subset of R^n and suppose that S \subset U is a flatly
embedded S^{n-1} which is null homotopic by a homotopy h: S x I --> U.
Let D be the bounded complementary domain of R^n - S so that S \cup D
is a closed n-ball (since S is flat). Note that if D is a subset of h(S
x I) we are done - so assume that h(S x I) misses a point of D. Since D
is not contained in U let r: U \to U-D be the radial retraction of S
\cup D - point onto S. Since S \cup D is a ball, there is a retraction
R of U onto S \cup D. It then follows that r R h : S x I --> S is a
null homotopy of the identity of S, which is a contradiction.

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