Topology / complex analysis / what
Dec. 6th, 2016 10:50 pm![[personal profile]](https://www.dreamwidth.org/img/silk/identity/user.png){kind=small}
jackFor a plot bunny (yes, really :)):
You have a multivalued function from a sphere onto "some surface", continuous everywhere except two points. (Or, equivalently, a function from "some surface" to the sphere, I guess?)
If you look at points on the surface which map onto the same point on the sphere, and connections between them of "paths" on the sphere (up to continuous deformation), I feel like they end up acting like the integers, where "+1" and "-1" correspond to a clockwise of anticlockwise circumnavigation. Or possibly some subset, a cyclic group of some finite order, if there are repeats. Is that right?
If you have *three* points, what can the relationship between the points look like? What about more?
I remember doing something like that but not what it's called.
I'm trying to put something like the shadows of amber onto a more concrete mathematical footing :)
You have a multivalued function from a sphere onto "some surface", continuous everywhere except two points. (Or, equivalently, a function from "some surface" to the sphere, I guess?)
If you look at points on the surface which map onto the same point on the sphere, and connections between them of "paths" on the sphere (up to continuous deformation), I feel like they end up acting like the integers, where "+1" and "-1" correspond to a clockwise of anticlockwise circumnavigation. Or possibly some subset, a cyclic group of some finite order, if there are repeats. Is that right?
If you have *three* points, what can the relationship between the points look like? What about more?
I remember doing something like that but not what it's called.
I'm trying to put something like the shadows of amber onto a more concrete mathematical footing :)
no subject
Date: 2016-12-07 12:07 pm (UTC)I was some of the way there. Free group is great. Basically, imagining this like a world map you can explore, every possible route takes you somewhere different. Or presumably, I could impose some order by choosing to identify some of the points, to make loops etc.
I need to think about that too, but want to work out what constraints I'd like. Like, something a bit like a normal map, where two different routes to the same point are *usually* commutative, or close to commutative, but there's enough variation that with some experimentation, you can find your way to completely different worlds.