Differential Forms and Smooth Spaces
Posted by Urs Schreiber
As we have discussed at length in the entry on Transgression, differential graded commutative algebras (DGCAs for short) have a useful relation to presheaves on open subsets of Euclidean spaces (smooth spaces, for short):
- every smooth space has a DGCA of differential forms on it;
- and every DGCA sits inside the algebra of differential forms of some smooth space .
On top of that, every finite dimensional DGCA can be regarded as the Chevalley-Eilenberg algebra of some Lie -algebroid , which linearizes some Lie -groupoid.
Here I want to talk about my expectation that
The smooth space associated to any Lie -algebroid this way plays the role of the space # of the Lie -groupoid integrating .
As motivation and plausibility consideration, recall that in rational homotopy theory and notably in the work of Getzler and Henriques, one obtains a simplicial space from by defining its collection of -simplices to be the collection of -valued forms on the standard -simplex…
… and notice that with the above and using the Yoneda lemma, we can equivalently think of this as the collection of -simplices in :
Mapping simplices into a smooth space is like computing its fundamental -groupoid , thought of as a Kan-complex. In simple situations, notably when is an ordinary Lie algebra, also the ordinary fundamental (1-)groupoid is of interest. And I think in this case, where the right hand side simply denotes the one-object groupoid with as its space of morphisms.
I am thinking, that hitting everything you see in sections 6 onwards in Lie -connections (blog, pdf, arXiv) with should have various nice consequences.
I want to better understand how nice exactly. That involves better understanding the properties of these functors in light of the above expectation.
All help is very much appreciated.
Posted at January 28, 2008 9:35 PM UTC
Re: Differential Forms and Smooth Spaces
One thing I want to do — and I’d be glad to do it with you — is identify a class of ‘nice’ smooth spaces such that their deRham cohomology matches the real cohomology of their underlying topological space.
Then, given a ‘nice’ smooth group , I would hope to show is a ‘nice’ smooth space, whose cohomology is related to that of in the way we expect.
For example, starting from we should be able to build up smooth spaces by iterating this construction.
I’ve got some of this working using Chen spaces. For example, I know how to form a Chen space from a Chen group . (I usually say ‘smooth’ instead of ‘Chen’, but I want to emphasize that there are various competing formalisms and I’m using this specific one.)
I haven’t made progress on isolating the ‘nice’ Chen spaces. But Mostow has an important paper on getting a smooth space for a smooth group , and working out its deRham cohomology. He uses a different formalism. Gajer has also done important work on making ’s into diffeological spaces. So, I’m sure some idea like this will work. The challenge is to get a setup where it all works smoothly.
(Pardon the pun.)