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wrote something at secondary characteristic class
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Thanks!
What’s the rule being exemplified there? Don’t put the anchor on the same line as the item?
@ Mike
Yes.
In the email which I have forwarded to the nForum last week, Jim said, among the rest, that the notion is slightly more general than the vanishing of curvature case. I have not quite undertood the remark but we should take it to the account. Jim said
The standard meaning of secondary (char classes or coh ops or even two centuries ago invaraints of algebraic forms) is of something that is not defined in general but only when some relation holds among primaries e.g Massey ops secondary coh ops
for char classes, curvature vanishing is an extreme example but what is more usual cf Chern-Simons is a char class that vanishes for dimensional reasons on a class of manifolds
(well CS is accounted for the Urs’s way, and I do not see it as a counterexample // Zoran)
By the way, with public provider of the mobile internet again problems in nlab (not nForum): Access denied. Your IP address, 95.168.108.22, was found on one or more DNSBL blocking list(s).
I had a hard time finding a reference that admits an actual precise definition of “secondary characteristic class”.
Moreover, even the semi-definitions that are in use are clearly not equivalent.
For instance in the school following Simons-Sullivans, authors use “secondary characteristic class” as a synonym for “Cheeger-Simons differential character”.
I wonder if Jim has a precise definition. On the other hand I am wondering if what he alludes to would be any different:
if we have a relation among characteristic classes, then clearly this means, I guess, that there is some combination of them that makes a vanishing characteristic class. I mean, if classes $c$ and $d$ satisgy a relatin like $c = d\smile d$ then clearly the class $c - d \smile d$ vanishes.
So is what Jim alludes to really more general?
Can we somehow reconcile the “vanishing” point of view on secondary classes with a relative/obstruction point of view ? I mean in other works you emphasised on Chern-Simons obstruction gerbe which happens when solving the lifting problem. But lifting problem can be at the level of Lie algebra reflected by not having the usual Chern-Weil theory, with usual Chevalley-Eilenberg complex, but rather the one with relative Chevalley-Eilenberg complex $CE(g,k)$, where we lift from the group $K$ integrating $k$ to group $G$ integrating $g$; and where the inclusion $K\subset G$ is a homotopy equivalence. This point of view on Chern-Simons via relative Lie cohomology is (if I understood the point), reviewed in Sec. 2 of Kontsevich’s article
The relative point of view for these classes seems to me more general than requiring some specific vanishings.
The relative point of view for these classes seems to me more general than requiring some specific vanishings.
Of course, relative means vanishing in a sense, but not in the sense of curvature. I am not saying that it is not equivalent, but it seems more direct to have it burried in the relative cocycle conditions.
Zoran,
this is a very good and important point. I should spend a bit more time thinking about it, but here is a first idea about what’s going on and how it fits into the general story.
Can we assume $K \hookrightarrow G$ to be a normal subgroup for the moment?
Then we have a crossed module of the same name, and thus a fiber sequence of $\infty$-groupoids
$\mathbf{B}K \to \mathbf{B}G \to \mathbf{B}(K \to G)$This says that given a $G$-bundle cocycle $X \stackrel{\simeq}{\leftarrow}C(U) \to \mathbf{B}G$ the obstruction to restricting it to an $H$-bundle cocycle is the composite 2-bundle cocycle
$X \stackrel{\simeq}{\leftarrow}C(U) \to \mathbf{B}G \to \mathbf{B}(K \to G)$
(so that’s the nonabelian “lifting”-gerbe). Indeed, as you can see in components, a trivialization of this $\mathbf{B}(K \to G)$-cocycle is precisely a section of the quotient $G/K$-bundle.
So possibly for the relative case one should consider $G$-bundles with connection, then send them further to $(K \to G)$-2-bundles with connection, and then, if these turn out to be flat, compute their Chern-Simons invariants as in $\infty$-Chern-Weil theory (which here just will boil down to the ordinary relative CW-theory).
Evidently, unless I am hallucinating, this would actually be a good example of the $\infty$-CW story. I should flesh it out.
Not sure right now what to do when $K$ is not normal in $G$, though.
Looks very nice for start. If one gets this entirely clear it may be very useful relating to a number of interesting cases.
Just back from dinner. Probably its different:
now I think we should start with the quotient K\G and then look at the action groupoid (K\G)//G. Then a reduction of the $G$-bundle is a lift of the cocycle through the projection (K\G)//G –> *//G = B G.
So then we should be looking for the invariant polynomials and Chern-Simons elements of the corresponding action Lie algebroid.
Just a quick remark, will get back to this tomorrow when I have a minute.
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Alexander Reznikov, Rationality of secondary classes, J. Differential Geom., Volume 43, Number 3 (1996), 674-692. (arXiv:dg-ga/9407007, euclid:jdg/1214458328)
Alexander Reznikov, All regulators of flat bundles are torsion, Annals of Mathematics Second Series, Vol. 141, No. 2 (Mar., 1995), pp. 373-386 (14 pages) (arXiv:dg-ga/9407006, jstor:2118525)
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