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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?

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1 Answer 1

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.

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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

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