Timeline for How to compute fundamental groups of closed surfaces without using Van-Kampen theorem?
Current License: CC BY-SA 4.0
8 events
| when toggle format | what | by | license | comment | |
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| Mar 16, 2020 at 20:12 | comment | added | Lee Mosher | The key words you want are "Poincare polygon theorem", or perhaps "Poincare fundamental polygon theorem". This question has some information. Entering those key words in my browser, this was the first hit. And, of course, look at the references in the answer of @MoisheKohan. | |
| Mar 16, 2020 at 4:45 | comment | added | Moishe Kohan | The first such computation is due to Poincare (Papers on Fuchsian functions), about 30 years before van Kampen's theorem. He was using fundamental polygons in the hyperbolic plane. A modern reference would be Beardon's book on discrete groups or, more generally, Ratcliffe's book on hyperbolic manifolds. | |
| Mar 16, 2020 at 1:57 | history | became hot network question | |||
| Mar 16, 2020 at 0:22 | answer | added | Nicholas Kuhn | timeline score: 6 | |
| Mar 15, 2020 at 23:07 | comment | added | Ryan Budney | Anything that uses a decomposition of the surface will be so close to Seifert-Van Kampen in spirit that I don't see how one could view it as different. Off the top of my head the only way I would imagine computing fundamental groups of surfaces without a decomposition of the surface would be by knowing the universal cover, and the group of covering transformations. This is effectively how the fundamental group of the circle is (typically) computed. | |
| Mar 15, 2020 at 22:24 | answer | added | Dmitri Pavlov | timeline score: 3 | |
| Mar 15, 2020 at 17:52 | comment | added | HJRW | I think it follows from the fact that the presentation complex of the presentation is the space whose fundamental group you are trying to compute. | |
| Mar 15, 2020 at 17:02 | history | asked | mathmetricgeometry | CC BY-SA 4.0 |