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]

I remember doing something like that but not what it's called.

I think the concept you're trying to retrieve is the fundamental group, which starts with the set of all loops in a space S connecting some distinguished base point b to itself (that is, continuous functions f:[0,1]→S with f(0)=f(1)=b, and f otherwise unrestricted in that it can self-intersect, revisit b repeatedly, whatever); then you endow it with the group operation of concatenation (given two paths f,g you take some 0<x<1 and form the path that squashes all of f into the interval [0,x] and g into [x,1]), and quotient out by the equivalence relation of homotopy (that is, any two paths that can be continuously deformed into each other within the space count as the same).

On your first example of the sphere with two points missing, you're right that the fundamental group is isomorphic to the integers under addition, with the equivalence class corresponding to some integer n being the set of all paths that loop n times anticlockwise (or −n times clockwise) around the axis between the two missing poles.

This is because you can basically retract the doubly-de-pointed sphere into a circle, and then it's obvious that it's only a matter of how many times you went round the circle, and that if you join together any two pieces of loop that go round the circle in opposite ways, they cancel each other out in the sense that you can homotopically unwind and get rid of any part where the combined path goes clockwise for a while and then anticlockwise back to the same place.

With three points missing, the fundamental group changes spectacularly, and becomes the free group on two generators, because now the space retracts to a figure of 8, and the interestingly distinct paths are of the form 'go round the left loop clockwise, then the right loop anticlockwise, then the left anticlockwise three times, then [finitely many more things along these lines]'. And if you concatenate two paths of that nature, you can't cancel the adjacent parts by the same homotopic unwinding technique unless they exactly undo each other, i.e. 'left anticlockwise' cancels against 'left clockwise' and ditto right, but 'left foo' and 'right bar' can't cancel at all. I.e. this is precisely the group generated by elements L and R with no relations between them.

With more than three points missing, you still get the free group on n generators for some n, and it's just a question of figuring out what n is. The sensible thing is again to retract your space S until it becomes some kind of wire-frame polyhedron skeleton, i.e. a graph G; then mentally expand the base point of your loops until it becomes a spanning tree T of that graph, and then the connected components of G \ T (being edges connecting some vertex of T to some other one without taking the approved route through T itself) each correspond to a generator of the fundamental group.

This is a construction which is also general enough to handle graphs other than those you get from a sphere with a few points subtracted, e.g. it lets you find out that the fundamental group of a wire-frame cube (say) has five generators (because, with 8 vertices, a spanning tree of it includes 7 of the 12 edges).

(And, of course, the original case of a doubly-de-pointed-sphere does drop out of that general construction as a special case, because the free group on one generator is the same group as (ℤ,+).)

[jack]

Yes, thank you, that's great.

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.