Timeline for Closed orientable surfaces have even Euler characteristic
Current License: CC BY-SA 4.0
5 events
| when toggle format | what | by | license | comment | |
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| Sep 12, 2018 at 23:10 | comment | added | Neil Strickland | Here's another way to say it: as $|D|$ is even, cycle decomposition shows that the parity of any permutation $\alpha$ is $|D/\alpha|\pmod{2}$. Thus $|D/\alpha\beta|=|D/\alpha|+|D/\beta|\pmod{2}$. Now take $\alpha=\sigma$ and $\beta=\chi$ so $\alpha\beta=\rho$, and note that $D/\rho\simeq V$ and $D/\sigma\simeq F$ and $D/\chi\simeq E$. This gives $|V|+|E|+|F|=0\pmod{2}$, as required. | |
| Sep 12, 2018 at 6:47 | comment | added | Gjergji Zaimi | @FrancescoPolizzi Orientability is equivalent to the existence of $\sigma$ as is explained in the question. | |
| Sep 12, 2018 at 6:31 | comment | added | Francesco Polizzi | Where do you use orientability? | |
| Sep 11, 2018 at 19:37 | vote | accept | Neil Strickland | ||
| Sep 11, 2018 at 18:35 | history | answered | Gjergji Zaimi | CC BY-SA 4.0 |