Tubular neighborhood of $X^k$ compact submanifold with normal bundle $\perp X$ trivial

Question

For $X^k\subset M^n$ compact submanifold with $\perp X$ trivial and set $S^k$ the $k$-sphere. Then there is a function $f:M^n\rightarrow S^k$ such that $X$ is the preimage for a regular value. My question is: the converse is true?

I guess (after answering) it should be rather $dim(X)=n-k$. – Daniel Valenzuela Nov 10 '14 at 07:39 — Daniel Valenzuela, Nov 10 '14 at 07:39

score 0 · Answer 1 · answered Nov 10 '14 at 07:37

0

Yes since you can pull back normal bundles and regular values tend to have trivial normal bundles.

answered Nov 10 '14 at 07:37

Daniel Valenzuela

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Tubular neighborhood of $X^k$ compact submanifold with normal bundle $\perp X$ trivial

1 Answers1