Timeline for Correct notion of "connected" for dga of bundle-valued forms
Current License: CC BY-SA 4.0
8 events
| when toggle format | what | by | license | comment | |
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| Jun 15, 2023 at 11:47 | comment | added | cheyne | Correct @Z.M but I'm trying to generalize some of the statements in bpb-us-e1.wpmucdn.com/sites.northwestern.edu/dist/c/2278/files/… , and this discussion for this MO post in particular is regarding Prop 2.4. | |
| Jun 15, 2023 at 5:16 | comment | added | Z. M | @DmitriPavlov The OP seems to be interested in the simply connected case, where there is no monodromy, and global sections of a vector bundle with an integrable connection could be identified with any fiber. | |
| Jun 15, 2023 at 1:29 | comment | added | cheyne | OK. My understanding is that in the case of real values forms, saying the dga is connected agrees with the case when the manifold is connected. For context, I am working on bundle valued iterated integrals and bar-like constructions on bundle valued forms. In this setting I want to “mod out” by a sub complex which is traditionally proven to be acyclic under some “connectedness” condition. I can say more if helpful. I appreciate the comments! | |
| Jun 15, 2023 at 1:11 | comment | added | Dmitri Pavlov | A point is as far from the generic case as possible, so there is no contradiction. It is hard to say more here unless a specific purpose for which you want a connectedness assumption is identified first. | |
| Jun 14, 2023 at 23:09 | comment | added | cheyne | Originally I agreed @DmitriPavlov but then you can see in my updated note that if $M$ is a point then it looks to me we should have $H^0 = End(F)$. | |
| Jun 14, 2023 at 23:08 | history | edited | cheyne | CC BY-SA 4.0 |
Update / additional note added.
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| Jun 14, 2023 at 13:07 | comment | added | Dmitri Pavlov | For a generic Ε,∇ the only ∇-flat sections of End(E) will be the ones given by multiplication by a real number. (The map that sends a ∇-flat section to its fiber at a fixed point is injective, but not necessarily surjective.) So the condition H^0≅R is quite natural. | |
| Jun 13, 2023 at 21:09 | history | asked | cheyne | CC BY-SA 4.0 |