In Diff, are the surjective submersions precisely the local-section-admitting maps? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-19T20:16:06Z http://mathoverflow.net/feeds/question/40309 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/40309/in-diff-are-the-surjective-submersions-precisely-the-local-section-admitting-map In Diff, are the surjective submersions precisely the local-section-admitting maps? David Roberts 2010-09-28T12:49:45Z 2010-09-28T18:15:35Z <p>Question as in title (Diff = category of smooth manifolds and smooth maps)</p> <p>I thought I'd convinced myself this is true, so this is just a sanity check.</p> <p>Also, what about for settings other than smooth manifolds? (like analytic manifolds, complex manifolds, or less differentiable - say, $C^2$ manifolds) </p> http://mathoverflow.net/questions/40309/in-diff-are-the-surjective-submersions-precisely-the-local-section-admitting-map/40312#40312 Answer by AndrĂ© Henriques for In Diff, are the surjective submersions precisely the local-section-admitting maps? AndrĂ© Henriques 2010-09-28T13:32:10Z 2010-09-28T13:32:10Z <p>There are two possible meanings for the sentence "<em>f</em> : <em>M</em> &rarr; <em>N</em> admits local sections", so let's first disambiguate.</p> <p><b>Meaning 1:</b> For every point of <em>N</em>, there exists a neighborhood of that points and a section from that neighborhood back to <em>M</em>.</p> <p>That's what people typically check in order to verify that, say, a map is a $G$-principal bundle.</p> <p><b>Meaning 2:</b> For every point <em>m</em> &isin; <em>M</em>, there exists a neighborhood of $f(m)$, and a section <em>s</em> from that neighborhood back to <em>M</em>, subject to the extra condition that $s(f(m))=m$.</p> <p>Clearly, you care about the <b><em>second</em></b> meaning of that sentence.</p> <hr> <p>It is correct that a map is a submersion (not necessarily surjective!) iff it admits local sections.</p> <p>If a map has local sections, then the maps on tangent spaces are sujective: that's just obvious.</p> <p>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 <em>M</em>, and apply the implicit function theorem.</p> <p><b>Note:</b> 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.</p> <p><b>Note:</b> For complex varieties, you don't have the implicit function theorem, so it doens't work. Counterexample: the map $z\mapsto z^2$ from &#8450;* to itself. The fix is to pass the the "&eacute;tale topology"... but that's another story.</p> http://mathoverflow.net/questions/40309/in-diff-are-the-surjective-submersions-precisely-the-local-section-admitting-map/40313#40313 Answer by Georges Elencwajg for In Diff, are the surjective submersions precisely the local-section-admitting maps? Georges Elencwajg 2010-09-28T13:32:19Z 2010-09-28T18:15:35Z <p>Dear David: yes!</p> <p>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 $x$.</p> <p>The other direction is <em>not</em> 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.</p> <p>However it is <strong>not</strong> 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.</p> <p><strong>Edited (later):</strong> 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.</p>