Local systems — what do you know about connections? May 28, 2009
Posted by David Speyer in Uncategorized.trackback
I’m working on the next local systems post, which will be about connections on vector bundles. It’s getting really long though. So, a question for my audience: Should I
(1) Assume you have already seen the definition of a connection, and of integrability (also known as flatness, also known as zero curvature) and skip directly to relating this concept to the other ideas I have discussed?
(2) Build connections from scratch, with motivation and definitions?
Don’t you wish every election featured such great choices? Ralph Nader would just go home and take up knitting!
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I am casting a vote for “…from scratch, with motivation and definitions…”.
I don’t think anyone would seriously object to the post being too long, except for you since you have to write it. So if you don’t mind the extra work it seems that people like me could get more out of it.
On the other hand, I have not read all the previous posts yet (but I plan to do so soon) so maybe my vote doesn’t count.
I’ll vote for from scratch as well. I’ve seen the definitions only once before and am not very familiar with all that stuff. . . this post might be the thing to firm it all up!
I vote for (2) as well
From scratch, please…
Well, one vote vs four.. I’m for the first option. Maybe you could just give a nice link for those who are interested? Anyway, a scratch won’t help much.
I was gonna vote for (1), thinking along the lines of A. Fonarev’s comment above (i.e., we could use a pointer/link towards the details, etc). But, personally, i don’t object to ‘post size’, in fact, quite the opposite: i enjoy reading well done articles/posts/newspieces/etc… it’s not about the length, but about a ‘well-done research’.
I’ve been enjoying the posts on Local Systems, all different approaches, and so on… good going! :-)
I vote (1).
Here’s a trivially-found long article already written.
I agree with Alex that I don’t think anyone would hold it against you if you included basic material on connections, and it’s possible you’d reach a broader audience that way. But if you’d prefer not going into that, or if it feels like too much of a digression, then I’d say just link to an already-written article and say what you really wanted to.
As for “Don’t you wish … Ralph Nader would just go home and take up knitting” — no, not really. I think he sometimes has useful things to say, and there should be opportunity for dissenting voices to be heard. You can always “change the channel”, after all!
Ummm, you do realize that your ellipsis completely change my meaning, right?
Since people are voting both ways I am just going to modify my original vote.
I don’t actually care about from scratch or definitions.
It is just hard to pass up someone writing a post that includes motivations.
I would like something from scratch with motivations (and definitions if you think that helps too), primarily because I have never been exposed to any Riemannian geometry, and all sources seem to presume you already at least presume you know the basic motivating questions and techniques of Riemannian geometry.
@ David: yes, I can see that it might, but then again I wasn’t sure what you meant. You might have meant wouldn’t it be great to have a voting scheme alternative to what we have now, or maybe that not having Nader on the ballot at all would have been a good choice, or something else. But still the last part carried a tinge of suggestion (to me) that you wish Nader would just go home and shut up. I’m sorry if I misread you, and I’m happy to let it go at that.
(2) please!
I’d vote for (1) and also vote for splitting the post once it gets too long.
David,
I think you may have set a false choice for yourself. I vote that you take the middle road, and use your judgement.
Of course, part of this is that I tend to despise the usual exposition of the theory of connections. I had to see it at least 4 times before it made sense to me (this part of a more general dislike of how differential geometry is exposited. It tends to be low tech in a way that’s confusing rather than helpful).
I’m really a lot happier thinking of it as a splitting of a central extension of Lie algebroids, but that’s probably just me.
As for Nader, I think that it was both a rather odd way for David to make his point, and an extremely odd way for Todd to misread that point.
On the other hand, the fact that it involved the mental image of Ralph Nader sitting by the fire and knitting still gives it more redeeming value than our “universal health care” discussion.
If you have a perspective or exposition on connections that is substantially different from the current wikipedia article (linked by Allen in comment 7), then I would be very interested to see it. Even though another treatment exists and is easy to find, this doesn’t imply that another write-up can’t be very helpful. For example, the introduction of the article mentions the intuitive idea of relating fibers over nearby points, but this idea is not mentioned again until much later (in particular the holonomy section), and not in a direct way. A treatment that brings the two perspectives closer could be a valuable contribution.
I gather you are all fond of how a local system is a representation of the fundamental groupoid Pi(X).
A more general connection (not necessarily flat) is similarly the same thing as a representation of a path groupoid that sits over the fundamental groupoid P_1(X).
Here is how this relates to the Lie algebroid picture that Ben is favoring:
given a G-principal bundle P –> X there is its Atiyah Lie groupoid sequence
Ad P –> Trans(P) –> X x X
When we pull this back to the fundamental groupoid, its splittings are flat connections on P. If we pull back further to the path groupoid, its splitting are all connections on P.
If we Lie-differentiate this, we get the Atiyah Lie algebroid sequence. Its splittings at the level of Lie algebroids are still flat connections on P. It splittings at the level of vector bundles are all connections, with their curvature being the obstruction to refine the splitting to one in Lie algebroids.
There is a way to say not-necessarily-flat connection in terms of Lie algebroids entirely. That requires passing to higher Lie algebroids but has the advantage that then it also takes care of higher connections. (This is described in articles with Jim Stasheff and Hisham Sati.)
I started writing s piece about “connection” here
[[connection on a bundle]]
There is more to do here, but I am running out of time. I do describe the main idea, the way I am thinking it should be presented (different from what Wikipedia has…) and point to literature which has all the details.
More later… (well, I am about to hop on a plane, so maybe quite a bit later…)
I have always had a sense of amusement about connections and curvatures. They seem to capture so much geometric intuition behind them yet they can be described completely algebraically.
I will be really interested if you want to present some novel viewpoint on this otherwise wikipedia article is fine enough.
i vote for 2)
I remember a reading earlier posts and wondering “aren’t the gadgets which lets you compare fibers called ‘connections’?”
option two (2) please. i believe in redundancy because you can slant things your way. emphasizing here, leaving out details there.
e.g. this is a connection. let X be the moduli space of connections on this 4-manifold
or this is a connection. this is some gnarly categorical way of defining connections that will be useful in our talk.
From scratch #1
Thanks