Timeline for Determine monodromy representation from local system
Current License: CC BY-SA 4.0
14 events
| when toggle format | what | by | license | comment | |
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| Sep 24, 2023 at 19:21 | comment | added | Moishe Kohan | Yes, Chech..... | |
| Sep 24, 2023 at 19:20 | comment | added | JackYo | @MoisheKohan: just to be sure that we are talking about the same construction: by nerve constructed from a cover you mean the Cech nerve (ncatlab.org/nlab/show/cover#definitions), right? (sorry, if it's cristal clear, I'm far from beeing an expert on field of simplicial techniques) | |
| Sep 24, 2023 at 19:08 | comment | added | Moishe Kohan | Later. But the direction is the opposite one: a cover determines its nerve, a cocycle defined via the cover defines a group homomorphism. | |
| Sep 24, 2023 at 18:54 | comment | added | JackYo | @MoisheKohan: Could you explain in a bit more details how to describe a cover from the nerve data - ie the simplicial model of classifying space - of the fundamental group as you suggest? | |
| Sep 24, 2023 at 18:53 | vote | accept | JackYo | ||
| Sep 24, 2023 at 7:05 | history | became hot network question | |||
| Sep 24, 2023 at 2:27 | comment | added | Moishe Kohan | The word "explicit" is ambiguous. "Algorithmic" is better, but then the input data has to be given in a suitable form. One way to define a flat bundle in a computable form is via a 1-cocycle for a sufficiently good finite open cover. The next issue is the fundamental group: one can describe a cover by its nerve. The 1-skeleton of the nerve will determine the group generators (once a maximal tree is chosen). Given all this, there is a relatively straightforward algorithm for computation of images of generators. | |
| Sep 24, 2023 at 1:09 | answer | added | R. van Dobben de Bruyn | timeline score: 5 | |
| Sep 24, 2023 at 0:45 | answer | added | Tom Goodwillie | timeline score: 3 | |
| Sep 24, 2023 at 0:44 | comment | added | Vik78 | Starting with a local system, can you define the map as follows? Take your fiber $V$ over a basepoint $x \in X$, pull it back via a local isomorphism $p$ to a point $\tilde{x} \in \tilde{X}$ over $x$, apply a deck transformation $g$, and then define the representation to act on $V$ via $\rho(g) = \pi_* \circ g \circ (p_*)^{-1}$. If you have a reasonably explicit description of your covering map I think you can make this pretty explicit. Perhaps you run into a similar problem by trying to explicitly describe the pullback of your local system to a local system on $\tilde{X}$ though | |
| Sep 23, 2023 at 23:53 | history | edited | JackYo | CC BY-SA 4.0 |
added 12 characters in body
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| Sep 23, 2023 at 23:49 | comment | added | LSpice |
Did you really mean $\operatorname{GL}_m(\mathcal C)$ at the end, and note $\operatorname{GL}_m(\mathbb C)$? \\ TeX note: \mathrm is not for "\rm in a math-mode environment", but for "math in \rm". For the latter, you want something like \textrm, although actually it's just \text. Note $\text{local systems}$ \text{local systems} vs. $\mathrm{local systems}$ \mathrm{local systems}. Similarly for \mathit, which in your case should have been \textit. I edited accordingly.
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| Sep 23, 2023 at 23:10 | history | edited | LSpice | CC BY-SA 4.0 |
TeX and proofreading
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| Sep 23, 2023 at 23:00 | history | asked | JackYo | CC BY-SA 4.0 |