Timeline for Formality of de Rham algebra for two-dimensional closed surfaces
Current License: CC BY-SA 3.0
10 events
| when toggle format | what | by | license | comment | |
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| Oct 7, 2011 at 15:27 | comment | added | pmnev | Dear Bruno, I did not mean to question formality of De Rham dga for a surface (I know that it holds due to a general theorem for Kaehler manifolds). I was rather asking whether formality holds in more strict sense, that you can find a dga morphism (not just any $A_\infty$) from cohomology to forms. To this I received a very clear answer from Robert Bryant, see below. | |
| Oct 7, 2011 at 14:50 | comment | added | Bruno V. | Dear Pasha, a dga algebra is formal iff there exists an $A__\infty$-quasi-isomorphism between it and its cohomology with induced (stric) algebra structure, not only an $A_\infty$-morphism. (Not this is just a typo, I am sure). | |
| Oct 4, 2011 at 13:11 | vote | accept | pmnev | ||
| Oct 4, 2011 at 12:48 | answer | added | Robert Bryant | timeline score: 14 | |
| Oct 4, 2011 at 0:47 | comment | added | José Figueroa-O'Farrill | @pmnev,algori: Thanks for your comments. I understand now what the question actually asks. | |
| Oct 4, 2011 at 0:24 | comment | added | algori | pmnev -- I think you are right. Moreover, I think one can get a genuine morphism of algebras at the expense of enlarging the target: namely, if one replaces the de Rham algebra with the cobar of its bar. | |
| Oct 4, 2011 at 0:20 | comment | added | algori | Jose -- this just shows that Sullivan's minimal model maps quasi-isomorphically to both algebras. | |
| Oct 4, 2011 at 0:08 | comment | added | pmnev | As far as I understand, formality just means that there exists some A_\infty morphism from cohomology to de Rham algebra. Generally it would have polylinear components. My question was whether one can find an A_\infty morphism which has only linear component. | |
| Oct 3, 2011 at 23:50 | comment | added | José Figueroa-O'Farrill | Does this not follow from the formality of compact Kähler manifolds, proved in Deligne, Pierre; Griffiths, Phillip A.; Morgan, John W.; Sullivan, Dennis (1975), "Real homotopy theory of Kähler manifolds", Inventiones Mathematicae 29 (3): 245–274, doi:10.1007/BF01389853, MR0382702 ? | |
| Oct 3, 2011 at 14:27 | history | asked | pmnev | CC BY-SA 3.0 |