Topology / complex analysis / what

[jack]

For 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 :)

Comments

[simont]

Really horrible afterthought:

There is another well-known kind of group I can easily bring to mind in which each individual generator has a small finite order but the group as a whole is ginormous and you need to combine those simple generators into very long words to get between an arbitrary pair of points.

Omit seven points from your sphere, and arrange that the six resulting generators of your worldwalking group correspond to quarter-turns of the six faces of a Rubik's cube :-)

[jack]

:)

[simont]

(I vaguely recall someone telling me once, on that subject, that there's a text adventure game set on the interior of a Rubik's cube – the rooms are organised in a 3×3 grid and there are some puzzle actions you can take to perform face turns...)