Can ∞-Chern-Weil theory provide an answer to this MO question:
I am wondering if there is a more general or abstract framework that allows one to define the Stiefel-Whitney classes in the spirit of Chern-Weil.
At least not in any immediate way. The reason is this:
-Chern-Weil theory (and ordinary Chern-Weil theory in special cases) reads in a characteristic class given by a morphism
of -Lie groups, and then produces from this a characteristic class with coefficients the differential forms with values in .
Now, an -valued differential form sends infinitesimal paths in some base to infinitesimal paths in . If is a discrete group then the only infinitesimal paths inside it are constants and hence there are no nontrivial differential forms with values in a discrete group.
So for instance for the real line divided by a discrete group, the corresponding differential forms are forms with values in the real numbers.
But for just a discrete group itself, the corresponding differential forms are trivial.
This is what the person asking the question alludes to: SW-classes have coefficients in and this necessarily vanishes in de Rham cohomology, where Chern-Weil theory could say something about it.
Now, if understood correctly, Chern-Weil theory does not actually forget any information about SW classes, because the refined CW homomorphisms remembers not just the differential forms, but also the bundle that they are curvature forms on. So in the refined CW homomorphism the image of an –principal bundle under an SW class is is a -principal -bundle representing that class, with an -connection that represents that class in de Rham cohomology. But that connection is necessarily flat and the image of the class in de Rham cohomology contains no information.
]]>Can -Chern-Weil theory provide an answer to this MO question:
]]>I am wondering if there is a more general or abstract framework that allows one to define the Stiefel-Whitney classes in the spirit of Chern-Weil.
I am working on finalizing some things. Today I went through the Motivation-section at infinity-Chern-Weil theory and polished it a bit more and added missing references.
]]>@Jim,
I hope this > works - if not, tell me how else to cut and paste
to get the
>
to work, you need to select ’Markdown’ below the comment box.
]]>Jim, we are just talking about the formal dual of a general Weil algebra.
]]>do you mean something more than g acting on g[1] by the adjoint actions shifted?
For an ordinary Lie algebra, nothing else is meant.
]]>Sorry, i wasn’t concentrating and then the new -notation led me astray.
We need to write , with the angular brackets. And then, yes, the diagram, for genuine -connections, is
]]>[edit: removed after some personal email conversation, see comment below]
]]>[edit: removed after some private email conversation, as being all caused by a stupid misunderstanding of notation, see comment below now]
]]>I’d rather have
where now inside plays exactly the same role of inside .
(by the way, is nothing but , but I prfer writing it as in the big diagram)
]]>Right, so a genuine connection (generally: genuine -connection) is characterized, in this language, precisely by the fact that its curvature forms have “no legs along the simplicial direction”.
So the diagram we want is
where now the left bottom morphism is just the gca-inclusion, not a dgca-morphism.
]]>Concerning what I write in #169, if the big diagram in #126 where
instead, then everything would be clear to me.
]]>how do you copy parts of previous to appear in a blue box?
Jim, a good trick to see (and copy) what others have done is to click the “Source” link on the top right of each comment.
This lets you even copy and paste the latex commands, diagrams, etc. Very useful. One of my favorite technical features of the n-Forum. I wish the n-Cafe had this feature.
]]>I think we have been taking this to private email for the moment.
But for the record:
Jim was talking about “fiber” in a different sense than Domenico and I was, and I think it is easy to see that the quasi-iso that seemed to materialize in above comments cannot exist. But maybe one can say something about useful isos on generators of cohomology rings.
]]>It seems to me that one would still need the lengthy discussion that i posted above
yes, sure. I meant that one had no more to wonder why to use that definition for : the simple answer is: because it is an extremely nice model for .
]]>Domenico,
concerning this remark:
if […(this is the cae)..] nothing else needs to be said!
It seems to me that one would still need the lengthy discussion that i posted above. But we’d be guaranteed that, in the words of this discussion, the nice choice of span would always exist.
]]>mmm.. and what about considering morphisms of Lie algebras (the source of the morphism is “in degree -1”) as the starting point ? so that
this way the obvious commutative diagram
would naturally induce the desired .
with Marco Manetti we considered a similar question in math.QA/0601312, but there the starting point was a pair of morphisms , seen as a tiny bit of a semicosimplicial object in dglas, whereas here we would rather be considering an augmented semicosimplicial object.
]]>now we have to go inside to recognize H(CE(g[1]) ) as invariant polynomials
That’s the crucial step.
]]>I must not have understood the question
Yeah, I just checked my email to remind me. That was January 2008. Re-reading it now, I see that you probably intended to tell me that it is a homotopy fiber sequence. But I understood that you were asking me if that was my question. Then I asked you to tell me how we would check this. And then you said you didn’t know and forwarded that question to Johannes Huebschmann. Who also didn’t give the answer!
I suppose it must have been a huge misunderstanding between us three. I am also thinking now this will have an easy answer and that I will hate myself for not having thought this through in the -case years ago.
how much of the following do you agree we know:
CE(g[1]) –> W(g) –> CE(g) is a fibration of dgcas
Let’s just get on the same page with terminology. is degreewise surjective, hence a fibration in the standard model structure on dgcas. Is that the meaning of “fibration” we are both talking about?
Okay, and then what do you mean by ? Is this that supposed to be the kernel of the map ? In that case I would call this the ideal generated by in . Is that what you mean by ?
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