|WikiProject Mathematics||(Rated Start-class, Mid-importance)|
This article does not actually define what the associated bundle construction is. 126.96.36.199 07:02, 4 Dec 2004 (UTC)
That's true enough: you need to take an open cover Ui and talk about the transition function data for transition on intersections; and then make the obseveration that the same G-valued data gives you many associated bundles. I suppose this page was awaiting proper definition of fiber bundle, and transition function.
Charles Matthews 16:08, 4 Dec 2004 (UTC)
Relation with subgroups
One very useful case is to take a subgroup of . Then an -bundle has an associated -bundle: this is trite for bundles, but looking at their sections it is essentially the induced representation construction, in a different light. This does suggest there will be some adjoint functors involved.
Complexifying a real vector bundle
One application is to complexifying a real vector bundle (as required to define Pontryagin classes, for example). If we have a real vector bundle , and want to create the associated bundle with complex vector space fibers, we should take and in that schematic.
Can't we just say what it is without all of the subterfuge?
I admit that I'm not all that familiar with the most general definition of an associated bundle. But in common parlance among my circles, an associated bundle is simply a bundle associated to a principal bundle by means of a fibre product. Earlier definitions (Steenrod, The topology of fibre bundles, Princeton University Press: 1950) disappointingly seem to propagate the misconception that associated bundles are some kind of mysterious concept. Now I may have missed some particular subtlety, but my understanding is that (using Steenrod's nomenclature) two bundles are associated if and only if they are associates of the same principal bundle. So I find the very idea of "associated bundles" to be a bit of a non sequitur. I don't mean to say that the article shouldn't exist, but I do believe that it can be clarified extensively. I moved a few of the vague sections into the talk page for now. Silly rabbit 07:53, 13 November 2005 (UTC)
- There is a small issue here (you may be right about squeamishness about principal bundles, actually). It's just that for a faithful action of group G on fiber F, going from the F-bundle to the G-bundle is reversible; therefore you have the choice of going from an F-bundle to an F′-bundle directly, or via the G-bundle and then 'reversing' to F′. This is no big deal for those who understand it all intuitively, but is the kind of point on which one should have mercy on the new student. Charles Matthews 09:14, 13 November 2005 (UTC)
- I, for one, find the principal bundle picture more intuitive. (Of course, I've been working with them for some time, so I am indeed already biased.) Perhaps we should compromise. The idea behind associated bundles is just that they have the same structure group and "the same" transition functions (i.e., cocycle) when changing coordinates. (These transition functions are only the same in the sense that they are given as the same G-valued functions on the coordinate overlaps.) Of course, G may act in different ways on the fibre of F and that of F′. I find this definition to be a bit disingenuous because it already slips the principal bundle in through the back door. But you're basically right. I can see how this definition through "subterfuge" can be more easily understood by someone who isn't yet comfortable with principal bundles. I'll do my best to preserve the spirit, if not the actual letter, of the article. Thanks for sorting me out, Silly rabbit 16:42, 13 November 2005 (UTC)
Export reduction of structure group to Principal Bundles
When I get the chance, I intend to export reduction of the structure group to principal bundles. (This is a self-note, as well as an indication of my intentions to other Wikipedists.) Silly rabbit 08:07, 13 November 2005 (UTC)
- Once I've done that, the rest appears passable. Silly rabbit 09:00, 13 November 2005 (UTC)