2
$\begingroup$

This is from Guillemin and Pollack chapter 2.

Theorem: If $X$ is any compact manifold with boundary, then there exists no smooth map $g:X\to \partial X$ with $\partial g:\partial X\to \partial X$ the identity.

The proof proceeds by contradiction: Suppose such a $g$ exists and let $z\in \partial X$ be a regular value. They claim this follows by Sard's theorem, but I don't see why? Couldn't the boundary have measure zero (it could even be empty)?

My second question: They also claim that the codimension of $g^{-1}(z)$ in $X$ is equal to the codimension of $z\in \partial X$, and so is $\dim X-1$, but why is this true?

$\endgroup$
5
  • $\begingroup$ The point 1) is an application of the generalization of Sard's theorem proved page $62$ : for any smooth map $f : X \to Y$, $X$ being a manifold possibly with boundary, then almost every point is a regular value of both $f$ and $f_{|\partial X}$. For the second point, this is also an application of the theorem abut transversality page $60$. $\endgroup$
    – user171326
    Apr 26, 2017 at 20:04
  • $\begingroup$ @N.H. thank you very much. if you would like to answer, I would be happy to accept, otherwise I would just delete this. $\endgroup$ Apr 26, 2017 at 20:20
  • $\begingroup$ Sure I can make an answer. In fact I never read the book by Guillemin and Pollack but it seems to me that it's the same content as Milnor's book on differentiable topology (probably with more details). $\endgroup$
    – user171326
    Apr 26, 2017 at 20:24
  • $\begingroup$ @N.H. yes we are using them in tandem. Milnor doesn't have much exposition haha. Anyway, thank you for your answer. It was an error to skip the transversality section obviously. $\endgroup$ Apr 26, 2017 at 20:28
  • 1
    $\begingroup$ Sure ! In fact, I'm glad the book you're reading is more detailed, since I was never sure to be able to fill rigorously the details... And now I see 50 pages where Milnor was writing 2 pages haha, but on the other hand it's a very pretty book ! $\endgroup$
    – user171326
    Apr 26, 2017 at 20:31

1 Answer 1

1
$\begingroup$

The point 1) is an application of the generalization of Sard's theorem proved page $62$ : for any smooth map $f : X \to Y$, $X$ being a manifold possibly with boundary, then almost every point is a regular value of both $f$ and $f_{|\partial X}$.

For the second point, this is also an application of the theorem concerning transversality page $60$.

$\endgroup$
3
  • $\begingroup$ There should be some condition on $Y$ there. For example, $\dim X\geq \dim Y$ or something like that. (I don't currently have access to the book, so I can't look it up.) $\endgroup$ Apr 26, 2017 at 20:33
  • $\begingroup$ @JasonDeVito : Are you sure ? If $\dim X < \dim Y$, then the image has measure zero, so almost every point is a regular value for $f$ and $f_{| \partial X}$. $\endgroup$
    – user171326
    Apr 26, 2017 at 20:39
  • 1
    $\begingroup$ Duh!. Objection withdawn and +1 from me. $\endgroup$ Apr 26, 2017 at 20:57

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .