In Diff, are the surjective submersions precisely the local-section-admitting maps? - MathOverflow

In Diff, are the surjective submersions precisely the local-section-admitting maps?

2010-09-28T12:49:45Z

Question as in title (Diff = category of smooth manifolds and smooth maps)

I thought I'd convinced myself this is true, so this is just a sanity check.

Also, what about for settings other than smooth manifolds? (like analytic manifolds, complex manifolds, or less differentiable - say, $C^2$ manifolds)

Answer by André Henriques for In Diff, are the surjective submersions precisely the local-section-admitting maps?

2010-09-28T13:32:10Z

There are two possible meanings for the sentence "f : M → N admits local sections", so let's first disambiguate.

Meaning 1: For every point of N, there exists a neighborhood of that points and a section from that neighborhood back to M.

That's what people typically check in order to verify that, say, a map is a $G$-principal bundle.

Meaning 2: For every point m ∈ M, there exists a neighborhood of $f(m)$, and a section s from that neighborhood back to M, subject to the extra condition that $s(f(m))=m$.

Clearly, you care about the second meaning of that sentence.

It is correct that a map is a submersion (not necessarily surjective!) iff it admits local sections.

If a map has local sections, then the maps on tangent spaces are sujective: that's just obvious.

Conversely, if a map is surjective at the level of tangent spaces, you first pick a local section of the maps of tangent spaces. Then, to finish the argument, you use the fact that any subspace of the tangent space $T_mM$ is the tangent space of a submanifold of M, and apply the implicit function theorem.

Note: if you care about infinite dimensional Banach manifolds, then the existence of a section for the map to tangent spaces needs to be assumed a separate condition. Indeed, it's not enough to assume that the map of tangent spaces is surjective, since it's not true that any surjective map of Banach spaces has a section.

Note: For complex varieties, you don't have the implicit function theorem, so it doens't work. Counterexample: the map $z\mapsto z^2$ from ℂ* to itself. The fix is to pass the the "étale topology"... but that's another story.

Answer by Georges Elencwajg for In Diff, are the surjective submersions precisely the local-section-admitting maps?

2010-09-28T13:32:19Z

Dear David: yes!

In one direction this is just the functoriality of tangent maps. Let $f:X\to Y$ be the morphism, $x$ a point in $X$ with image $y\in Y$ and $g:V\to X$ a local section. From $f \circ g=Id_V$ you get $f_{\ast x} \circ g_{\ast y}=Id_{\ast y}$ and this implies that $f_{\ast x}$ is surjective i.e. that f is a submersion at $x$.

The other direction is not formal and depends on a theorem giving the local form of a submersion: this is much harder and is equivalent to the implicit function theorem or the local diffeomorphism theorm. It is true in the category of $C^k-$ manifolds, $k\geq 1$, and in that of real or complex analytic manifolds.

However it is not true in an algebraic geometry context. For example the squaring map $\mathbb C\to \mathbb C:z\mapsto z^2$ is a surjective submersion but has no local (in the Zariski sense) algebraic (= rational) section.To remedy this, Grothendieck introduced a new branch in Algebraic Geometry called Etale Topology, and more generally Grothendieck Topologies.

Edited (later): I hadn't defined "admitting local sections". Just as Tim observes in his comment, the answer "yes" is only correct with the understanding that through every $x\in X$ there passes a section defined in a neighbourhood of $y=f(x)$. This is also the " Meaning 2" in André's post, who quite judiciously chooses it as the relevant one.