Timeline for differential form of charge for pi_4(S^3) or pi_4(S^2)
Current License: CC BY-SA 2.5
12 events
| when toggle format | what | by | license | comment | |
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| Nov 16, 2010 at 7:42 | history | edited | Ivan Zhogin | CC BY-SA 2.5 |
mistype corrected
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| Nov 15, 2010 at 11:40 | comment | added | Ivan Zhogin | @ José Figueroa-O'Farrill - Yes, Skyrme model and Faddeev model are the close issue. @ David Roberts - thanks, Cheeger-Simons gives many Google-links, but.. it's not so easy. Perhaps it's better: to consider the subset of axi-symmetrical mappings: the set of orbits is 3-ball and the stationary points (where $f(h)$ commutes with the symmetry, $g=e^{i\alpha/2}$) as its boundary, $S^2$; so we have a diad (relative) mappings $(D^3, S^2) \to (S^3,S^1)$, which leads to the relative (or diad) homotopy group $\pi_3(S^3;S^1)$; to write 3-form of topological quasi-charge; to do something else :-) | |
| Nov 15, 2010 at 6:14 | comment | added | Ryan Budney | Try to look for Bott and Tu via Google. Homotopy groups of spheres are torsion outside of $\pi_n S^n$ and $\pi_{4n-1}S^{2n}$ so $\pi_4 S^3$ isn't detectable by standard differential form technology -- things like pull-backs, wedge products, integration, etc. | |
| Nov 15, 2010 at 6:14 | comment | added | David Roberts♦ | @Ivan - the book jc mentions is still in print, but here is a Google books link: books.google.com.au/… | |
| Nov 15, 2010 at 6:11 | comment | added | David Roberts♦ | You may want to look at differential characters a la Cheeger-Simons. They have forms as an ingredient, and can detect torsion. I'm not sure if they can see the classes you describe though. This may not give the answer, but point you in roughly the right direction. | |
| Nov 15, 2010 at 6:07 | comment | added | Ivan Zhogin | @ jc: Yes, the question is how to calculate, to prove that the map is non-trivial (with some integration, not only a framed manifold). 2. No, I have not. Have you any link on this book? Thanks. | |
| Nov 15, 2010 at 5:58 | history | edited | Ivan Zhogin | CC BY-SA 2.5 |
more detailed question; edited body
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| Nov 13, 2010 at 23:19 | comment | added | José Figueroa-O'Farrill | @Daniel: actually rational homotopy (by Sullivan,...), but your objection stands. I don't know how to probe a torsion class via differential forms. | |
| Nov 13, 2010 at 22:15 | comment | added | Daniel Pomerleano | How is this ever going to work? Don't differential forms only detect real homotopy? | |
| Nov 13, 2010 at 20:35 | comment | added | j.c. | I think Ivan Zhogin is after some differential form similar to the 3-form one can integrate on $S^3$ to get the Hopf invariant of a mapping. So he'd want a construction of differential forms on $S^4$ from maps from $S^4$ to $S^3$ and maps from $S^4$ to $S^2$, which when integrated yielded topological invariants (mod 2, say). Ivan, have you looked at Bott and Tu's book on Differential forms in Algebraic Topology? | |
| Nov 13, 2010 at 14:43 | comment | added | José Figueroa-O'Farrill | I'm not sure I understand this question. Where does the differential form live? Or are you asking for a map (à la Skyrme, say) $\mathbb{R}^5 \to S^3$, such that its restriction to the sphere at infinity gives the generator of $\pi_4(S^3)$? | |
| Nov 13, 2010 at 12:00 | history | asked | Ivan Zhogin | CC BY-SA 2.5 |