Cross sections in bundles and principal G-bundles - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-25T10:46:31Z http://mathoverflow.net/feeds/question/100584 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/100584/cross-sections-in-bundles-and-principal-g-bundles Cross sections in bundles and principal G-bundles Shlomi A 2012-06-25T11:28:39Z 2012-06-27T13:03:51Z <p>A principal $G$-bundle has a cross section iff it is trivial (e.g. Husemoller's Fibre Bundles, 3rd ed., 8.3 in chapter 4).</p> <p>A principal $G$-bundle is in particular a fiber bundle with fiber $G$.</p> <p>My question: does there exist a group $G$ and a <strong>non-trivial</strong> principal $G$-bundle $p:E\rightarrow B$ that <em>does</em> have a cross section when considered as a mere fiber-bundle?</p> <p>If so, I would be glad to see a simple example. Thanks!</p> http://mathoverflow.net/questions/100584/cross-sections-in-bundles-and-principal-g-bundles/100706#100706 Answer by Oldřich Spáčil for Cross sections in bundles and principal G-bundles Oldřich Spáčil 2012-06-26T17:56:41Z 2012-06-27T13:03:51Z <p>Let $p\colon E \to B$ be a principal $G$-bundle and $s\colon B \to E$ a global section. Then the map $F\colon B\times G \to E$, $F(x,g) = s(x)\cdot g$ is a global trivialization of $E$. Here the dot denotes the right action of the group $G$. The map $F$ is surjective, because the action is transitive on fibres, and it is injective because the action is free. Continuity/smoothness is the same as the continuity/smoothness of your section $s$ and the action of the group $G$.</p> <p>There's no special requierement in the definition of a section of a principal $G$-bundle, it's still the section of the underlying fibre bundle. But once you have a global section, you just use the $G$-action to trivialize the bundle.</p>