Timeline for Obstruction Theory for Vector Bundles and Connections
Current License: CC BY-SA 4.0
12 events
| when toggle format | what | by | license | comment | |
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| May 10, 2018 at 1:58 | comment | added | user40276 | Don't you want something like a 2-cocycle of the Cech-Deligne complex which is flat? Or a flat bundle gerbe with connection... | |
| May 9, 2018 at 20:38 | history | edited | S. carmeli | CC BY-SA 4.0 |
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| May 9, 2018 at 0:04 | comment | added | John Klein | @ChrisSchommer-Pries You're 100% correct. The difference is this: if $G$ is the topological structure group of the bundle and $G^\delta$ is $G$ with the discrete topology, then flatness is the same as asserting the bundle, which is classified by a map $X\to BG$ lifts up to homotopy through $BG^\delta$. If it lifts, then a choice of lift $X\to BG^\delta$ factors through $B\pi= P_{\le 1}X$, where $\pi$ is the fundamental group of $X$. This is demanding more than having the map $X\to BG$ factor through $B\pi$. | |
| May 8, 2018 at 14:44 | comment | added | Chris Schommer-Pries | I think being a flat bundle is a stronger property than being pulled back from $P_{\leq 1}X$. For example take $X$ to be the torus (or any genus $g$ surface with $g\geq1$). Then $X = P_{\leq 1}X$ agrees with its Postnikov truncation. So every vector bundle is a pulled-back bundle. However line bundles corresponding to non-trivial elements of $H^2(X, \mathbb{Z}) =\mathbb{Z}$ will have non-trivial first Chern class, and hence are not flat. | |
| May 8, 2018 at 5:29 | history | edited | Ben McKay | CC BY-SA 4.0 |
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| May 8, 2018 at 1:27 | answer | added | John Klein | timeline score: 5 | |
| May 4, 2018 at 21:37 | comment | added | John Klein | Saal Hardali: Let $E\to \Bbb CP^1$ be the canonical line bundle. This is not flat. $ \Bbb CP^1 = S^2$ and the bundle is classified by the inclusion map $S^2 \to BU(1) = \Bbb CP^\infty$. Note that this map is the map from $S^2$ to its second Postnikov section, so we get an example of what you want. | |
| May 4, 2018 at 9:15 | comment | added | Saal Hardali | ... btw do you know any example of a non flat bundle on a manifold which is pulled back from some postnikov section? (i'm not implying that finding something like this is easier than the question - just interested). | |
| May 4, 2018 at 9:10 | comment | added | Saal Hardali | That's not what I said. I said that being trivial on connected components (i.e. being a pullback of something on $P_{\le 0}X$) is not detectable from the curvature form alone and that there's a fix for this when you use differential characters. I have no idea about higher postnikov section and was just commenting about the "trivial case" being not so trivial here... | |
| May 4, 2018 at 8:01 | comment | added | S. carmeli | Well, I guess you are arguing against the statement that flatness is equivalent to reduction to the foundamental groupoid. It sounds strange given that representations of the foundamental group and flat bundles are equivalent categories. Anyway, flatness do imply reduction to the foundamental groupoid, and my question is if we can impose weaker natural conditions on the connection that will ensure reduction to the second Postnikov piece. I don't require equivalence here. | |
| May 4, 2018 at 6:06 | comment | added | Saal Hardali | Already for line bundles and the $0$-th postinkov section this is nontrivial. The data of a line bundle is equivalent to giving the integral first chern class. Modulo torsion this is the curvature form. But if the chern class is torsion then the bundle is flat but potentially non trivial. Differential characters can help you here. For line bundles there's a cheeger simons character which classifies line bundles with connections modulo gauge equivalence. This solves the case of line bundles for the 0th postnikov section at least. | |
| May 2, 2018 at 14:50 | history | asked | S. carmeli | CC BY-SA 4.0 |