Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

Let $X$ be a smooth manifold and $Y$ be a closed submanifold. Then there exists a neighbourhood $U$ of $Y$ in $X$ such that $Y$ is a deformation retract of $U$ right?

I can only find (stronger forms of) this in literature under the assumption that $Y$ is compact, which I don't think is necessary for the above statement. So what would be a reference?

share|cite|improve this question
up vote 5 down vote accepted

Since you just want a reference request: see Theorem III.2.2, Corollary III.2.3, and the remark after Definition III.2.4 of Kosinski's Differential Manifolds.

share|cite|improve this answer
Thanks for the reference. I am a bit confused. Does Kosinski say that in my situation $U$ can even be choosen diffeomorphic to the normalbundle of $Y$ in $X$? – Jan Mar 14 '12 at 18:55
@Jan: I believe that is exactly what Kosinski says. (Sorry, I don't have the book with me right now, so am going by memory, but when I looked yesterday I think he defined the tubular neighborhood in terms of being diffeomorphic to normal bundle.) – Willie Wong Mar 15 '12 at 9:14
Thank you again! – Jan May 15 '12 at 7:38

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.