Timeline for Symplectisation as a functor between appropriate categories
Current License: CC BY-SA 4.0
10 events
| when toggle format | what | by | license | comment | |
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| Aug 3, 2019 at 16:28 | history | edited | Peter LeFanu Lumsdaine | CC BY-SA 4.0 |
made link to paper human-readable
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| Jul 28, 2019 at 23:57 | vote | accept | TheGeekGreek | ||
| Jul 28, 2019 at 21:24 | comment | added | Heinz Doofenschmirtz | @TheGeekGreek You are right. I see now, where the sign issue arises. Take a contact manifold $(M,\alpha)$, then $d(t\alpha)$ is a symplectic structure on $M\times\mathbb{R}^\times $. For a contactopmorphism $F\colon M\to\tilde M$ ($F^*\tilde\alpha=f \alpha$ ), then the map $S(F)(x,t)=(F(x),\frac{t}{f})$ is a symplectomorphism. "Your" symplectization basically chooses the open subset with positive reals, but a morphism with with negative $f$ doesn't preserve this choice. | |
| Jul 28, 2019 at 20:11 | comment | added | TheGeekGreek | @HeinzDoofenschmirtz Yes, but using that exterior derivatives and pullbacks commute, we have that $$S(F)^*d(e^t \widetilde{\alpha}) = d(S(F)^*(e^t\widetilde{\alpha})) = d(e^te^{-\log|f|}F^*\widetilde{\alpha}) = d(e^t |f|^{-1}f\alpha)$$ Also if I compute it straight forward, we get the quotient $f/|f|$ in the end. | |
| Jul 28, 2019 at 19:41 | comment | added | Heinz Doofenschmirtz | @TheGeekGreek I think you made a mistake. Note that you have $d\log(|f|)=\frac{df}{f}$ for all non-vanishing functions $f$. | |
| Jul 28, 2019 at 19:38 | comment | added | Heinz Doofenschmirtz | @JoséFigueroa-O'Farrill thanks for the suggestion, I already edited my post. | |
| Jul 28, 2019 at 19:31 | history | edited | Heinz Doofenschmirtz | CC BY-SA 4.0 |
deleted 4 characters in body
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| Jul 28, 2019 at 19:19 | comment | added | TheGeekGreek | I also thought about modifying the $S(F)$ in the way you did, but unfortunately, at least as far as I can tell, it doesn't work since $$S(F)^*d(e^t\widetilde{\alpha}) = d(e^t \operatorname{sgn}(f) \alpha) \neq d(e^t\alpha)$$ in general. Thank you for the suggested paper! I will check it out. | |
| Jul 28, 2019 at 19:16 | comment | added | José Figueroa-O'Farrill | I think it's friendlier to link to the abstract instead of (or as well as) the PDF. People are sometimes on slow connections and they may wish to see what the paper is about before downloading. | |
| Jul 28, 2019 at 17:07 | history | answered | Heinz Doofenschmirtz | CC BY-SA 4.0 |