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Studying analysis on manifolds, I have found, in the proof of the existence of tubular neighborhoods, a reference to theorem 3.1.2 in "Topologie algebrique et theorie des faisceaux" of Godement. Without going through the machinery of the sheaves, at least now, is it possible to bypass the Godement's result? And, if yes, what is an accessible (not sheaf-theoretic) route?

This is the initial setting: $J:N\rightarrow M$ is a smooth embedding. $\pi:E\rightarrow N$ is a vector bundle, and $s_0:N\rightarrow E$ is the zero section of $\pi$. $\psi:U\rightarrow M$ is a smooth map from a neighborhood $U$ of $s_0(N)$ in $E.$ $\psi$ is a local diffeomorphism in each point of $s_0(N),$ and $\psi\circ s_0=j$.

At this point there is a reference to the argument of Godement in order to prove that: (*)There exists a neighborhood $V$ of $s_0(N)$ in $U$ such that $\psi|_V$ is a diffeomorphism.

What is the argument (differential geometric, not sheaf-theoretic) in order to conclude (*)?

Added by Mariano: I now have a copy in my hands. Theorem 3.1.2 reads (my translation):

Let $$0\to\mathscr L'\to \mathscr L\to\mathscr L''\to0$$ be a short exact sequence of sheaves of abelian groups. If $\mathscr L'$ is flasque, then for all open sets $U$ there is a short exact sequence $$0\to\mathscr L'(U)\to \mathscr L(U)\to\mathscr L''(U)\to0$$

He remarks that we therefore have a short exact sequence of pre-sheaves.

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If you told us what the result says... –  Mariano Suárez-Alvarez Feb 8 '11 at 18:22
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You use lots of \mapstos in weird places! :) –  Mariano Suárez-Alvarez Feb 8 '11 at 20:05
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A differential-geometric proof is available in Guillemin and Pollack's "Differential Topology". It's an exercise, but they set you up so that it's relatively straight-forward. The exercise is broken into two parts, 1st for manifolds in Euclidean space, then for submanifolds of manifolds. –  Ryan Budney Feb 9 '11 at 19:35
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It's also written out nicely in Milnor-Stasheff's Characteristic Classes (and probably most other texts on differential geometry topology). –  Dave Anderson Feb 9 '11 at 19:38
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Right, Hirsch's "Differential Topology" would be another source. Kosinski's "Differential Manifolds", Bredon's "Geometry and Topology", etc. –  Ryan Budney Feb 9 '11 at 19:59

2 Answers 2

up vote 12 down vote accepted

In the finite-dimensional setting, it's possible to construct tubular neighborhoods without anything like Godement's lemma. Many sources simply rely on a point-set topology argument that's based on the same idea as Godement's lemma (to be precise, I'm talking about the argument on p. 109 of Lang's book Differential and Riemannian Manifolds, which he says follows Godement). I'll explain another approach.

The idea is to use a Riemannian metric on the manifold $M$, which also induces a Riemannian metric on $TM$ (viewed as a manifold in its own right). The geodesic distance then gives a (topological) metric on $TM$. If $Y\subset M$ is a (not necessarily closed) submanifold, then a simple metric geometry argument can then be used to find a neighborhood of the zero section of $N(Y)$ (thought of as the perpendicular complement of $TY$ inside $TM$) on which the exponential map is injective. The key fact about the exponential map $f$ is that every point in the zero section of $N(Y)$ has a neighborhood on which $f$ is a diffeomorphism onto an open subset of $M$. (Edit: Note that in the finite dimensional setting, $N(Y)$ is automatically a locally trivial vector bundle. This does not seem to be the case for arbitrary infinite dimensional Riemannian manifolds, as discussed here: Orthogonal complements in Hilbert bundles. Hence the discussion that follows does not work in as great generality as arguments based on Godement's lemma.)

The general metric geometry fact is this:

Consider a metric space $T$ and subspaces $X, Y$, and $D$ such that $Y \subset X$ and $Y\subset D$. (Think: $T = TM$, $X$ is the zero section of $TM$, $Y$ is a submanifold of $M$, and $D$ is the domain of the exponential map, lying inside $NY$.) Let $f: D\to X$ be a continuous map that restricts to the identity on $Y$ (think: $f$ is the exponential map). Assume further that for each $y\in Y$ there exists $\epsilon_y >0$ such that $f$ restricted to $B_{\epsilon (y)} (y, D) = \{z\in D \,:\, d(z,y) < \epsilon(y)\}$ is a homeomorphism onto an open subset of $X$. Then there exists a subspace $D'$, open in $D$, on which $f$ is injective.

Proof. For each $y\in Y$, $f(B_{\epsilon (y)/2} (y, D))$ is open in $X$, hence contains $B_{\epsilon'(y)} (y, X)$, for some $\epsilon'_y < \epsilon_y/4$ (remember that $f(y) = y$). Now consider the inverse image $Z_y$ of $B_{\epsilon'_y} (y, X)$ under the restriction of $f$ to $B_{\epsilon (y)/2} (y, D)$. Since $f$ is a homeomorphism when restricted to this ball, $Z_y$ is open as a subset of $D$. Now I claim that $f$ is injective on $D' = \bigcup_{y\in Y} Z_y$. Say $f(z_1) = f(z_2) = y_0$ with $z_1 \in Z_{y_1}$ and $z_2\in Z_{y_2}$, and assume $\epsilon_{y_1} \geq \epsilon_{y_2}$. Then we have $$d(z_2, y_1) \leq d(z_2, y_2) + d(y_2, y_0) + d(y_0, y_1) < \epsilon_{y_2}/2 + \epsilon'_{y_2} + \epsilon'_{y_1} $$ $$< \epsilon_{y_1}/2 + \epsilon_{y_2}/4 + \epsilon_{y_1}/4 \leq \epsilon_{y_1}$$ (for the second inequality, note that by definition, $y_0 = f(z_i) \in f(Z_{y_i}) \subset B_{\epsilon'_{y_i}} (y_i, X)$ for $i=1, 2$). So $z_2$ and $z_1$ both lie in $B_{\epsilon_{y_1}} (y_1, D)$, and since $f$ is injective on this ball we have $z_1 = z_2$.

This argument is useful for the construction of equivariant tubular neighborhoods in certain infinite-dimensional settings. See http://arxiv.org/abs/1006.0063; I just updated it so I guess the new version will show up tomorrow. The equivariant version of the above argument is in Proposition 2.3 or 2.4, depending on the version.

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Dear Dan Ramras This argument was just what I was searching for! Thank you. –  Giuseppe Tortorella Mar 11 '11 at 6:45

Yes, it's possible. And this answer is serious!

If the result you want to apply is (say) about "flasque sheaves", but you're working about "indefinitely differentiable functions", then it will be easy to explain the proof in your specific setting without diving in the whole theory of sheaves.

That being said, I guess it will be possible to actually explain the proof you want... when you'll have explained exactly which result you want!

EDIT: The result you ask about is just some kind of implicit function application ; in fact the french wikipedia even has a full page on this... with no english equivalent apparently.

EDIT 2: Ok, third try at answering the question ; this result on flasque sheaves is exercise 1.16(b), p67 of "Algebraic Geometry" by R.Hartshorne. You'll find the basic theory of sheaves in H.Cartan's lecture (page 6 has half the result -- the part which doesn't need the flasque hypothesis) ; the second part of the lecture has the other half, the theorem page 5 -- and you'll notice the way he defines fine sheaves (page 1) is asking for partitions of unity -- which are available in your particular setting!

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I have inserted the initial setting, and the conclusion borrowed appealing to Godement. In the book I am reading there is not the statement of Theorem 3.1.2 from Godement. –  Giuseppe Tortorella Feb 8 '11 at 19:39
    
I have recovered a restatement of the result of Godement. –  Giuseppe Tortorella Feb 8 '11 at 19:52
    
The straightening out theorem is only local. My problem is about the existence of an inverse of $\psi$ on a whole neighborhood of the zero section $s_0(N)$, not only around any one of its points. –  Giuseppe Tortorella Feb 8 '11 at 21:00
    
If you can do it around any point in the zero section, it implies you can do it around the zero section. –  Deane Yang Feb 9 '11 at 19:43
    
Dear Deane Yang. I have to understand better what are the difficulties I have met, even reading differential topology texts, and eventually rewrite a question. Excuse me. –  Giuseppe Tortorella Feb 9 '11 at 21:23

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