From a connection 1-form on $M$, I can construct a parallel transport from which in turn I can construct a classifying map $M \to BG$. Is there a construction of such a classifying map directly from a connection 1-form?
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$\begingroup$ @JimStasheff, I'm slightly confused: To talk about a connection 1-form on $M$ you will have some principal G-bundle $P\to M$ to work over, and to such a bundle there is a unique classifying map $\phi:M\to BG$ such that $P=\phi^*EG$. $\endgroup$– Chris GerigCommented Nov 25, 2013 at 21:30
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$\begingroup$ @ChrisGerig: of course the classifying map is only unique up to homotopy. If one has a particular model for BG in mind (e.g. a Grassmannian) then it may be helpful to have a construction that is more canonical. For instance one might ask if there is a continuous a map from the affine space of 1-forms to the mapping space Map(M, BG). Of course one could just take a constant map landing at a classifying map for the bundle in question, so I guess I'm not sure what additional properties to ask for. $\endgroup$– Dan RamrasCommented Nov 26, 2013 at 5:51
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$\begingroup$ Chris, I was trying to talk as a physicist ;-): A connection 1-form A gives a covariant derivative d+[A, ]. Then I was thinking of the model of BG as the realization of the nerve of G. For now I would settle for the following: (how) does a connection 1-form determine transition functions for the bundle. $\endgroup$– Jim StasheffCommented Nov 27, 2013 at 19:56
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1 Answer
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17.11 of
constructs transition functions for the bundle (in step 5 of the proof), for any complete Ehresmann connection whose holonomy Lie algebra is finite dimensional.
answered Nov 28, 2013 at 20:24
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$\begingroup$ Thanks Peter Exactly what I want, at least under the conditions you state. No access to a library until next week. Any chance it's also in one of your papers available on line? $\endgroup$ Commented Dec 1, 2013 at 0:44
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$\begingroup$ Just click on it. I gave a link to the pdf file. $\endgroup$ Commented Dec 1, 2013 at 13:20