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.

[simont]

But I'll be fascinated to see what this has to do with Amber :-)

[jack]

I didn't start thinking about Amber, that analogy occurred to me later.

I was pondering, suppose going all the way round the earth leads you into a slightly-parallel world. Like, you go to America westwards, and you're in the universe with childrens books containing "Berenstain bears", and "Jim the cabin boy" from Pugwash. But if you go eastwards, you end up in the world with "Berenstein bears" and "Roger the cabin boy". Hence, everyone on the internet is terminally confused which is which because usually internet connections are routed the shortest route, but not always. And if you go all the way round, you may find the physical evidence in your house doesn't _quite_ match up with your memories.

(99.9% of the time, parallel you goes the same way and moves one MORE universe over, and meets slight additional differences. In a minority of worlds, the differences happen to include something that changes whether you take the trip, and you meet yourself :))

And once is a *small* change, the sort of stuff you can pass off. But if you go far enough along the manifold, you reach noticeably different parallel worlds and eventually sci-fi stuff.

But I wanted to make sure I understood the implications of that. Eg. that there's a discontinuity at the N pole, where you could run round and round it and go quite a distance into the parallel worlds.

But that seemed insufficient, like, you can't put all possible parallel worlds into a linear sequence. Hence, the "if there's three points, say N pole, S pole, bermuda triangle". (Or the points vary in different parallel worlds.)

And it occurred to me, it's similar to Amber (or DWJ's Homewardbounders). Except Amber sensibly doesn't go into details about which worlds connect to which. They have to be "close enough", but where you are the world doesn't matter much, as evinced by Corwin worldwalking by going in circles, and each world connects to plenty of others, and you choose which by using your amber-heritage.

But there are hints of a pattern, the traps in the first book imply that there are only so many possible routes, and the people in the Amber navy and trade ships etc imply that some of the time, there are semi-fixed routes you need less Amber heritage to traverse. And I wonder, how those fit together.

[simont]

Eg. that there's a discontinuity at the N pole, where you could run round and round it and go quite a distance into the parallel worlds.

I don't quite know whether you think that's a bug or a feature – of course it would depend on what kind of story you're setting in the result of this worldbuilding. (A game-breaking exploit like 'run round exact point location of north pole 1000 times in an hour' might make some kinds of story too easy because it would provide a trivial solution to problems that the plot needs to be a big deal, but on the other hand in a different kind of story the process of hypothesising, confirming, finding and exploiting that loophole might be the whole point of the plot.)

But if you think it's a bug, then here's a possible fix you might like.

In my other comment I mention that the fundamental group of basically anything in the class of spaces you're talking about is the free group on some finite set of generators, i.e. any word formed out of those generators and their inverses gives a group element, and the only distinct words that represent the same group element are due to cases where a generator appears right next to its inverse – after you cancel those obvious redundancies to reduce a word to 'lowest terms', the remaining set of words are all distinct elements of the group.

But any other finitely generated group arises as a quotient of a free group, by imposing additional relations between the group elements. And the quotient map is a homomorphism, i.e. behaves sensibly and consistently and respects the composition law. So you could easily rule that the actual group of all parallel worlds was not actually the free group, but merely its quotient by some set of relations of your choice. And a nice choice might be to assign each individual group generator a finite order (i.e. for each generator g, add some rule that says g^2=e or g^3=e or g^99=e or some such), but without adding any relations between the generators.

Result: you still have an infinite branching tree of parallel worlds available if you're prepared to do a lot of arduous travelling back and forth between 'poles' of the world, but no matter where you currently are in the group, running round and round the same 'pole' of the world – whichever pole it might be – only cycles you through some small finite set of worlds.

[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...)

[jack]

Oh, cool.

I wasn't sure between "bug" and "feature" :)

My first thought was, if you have something like this, there must be points of discontinuity, and what are they physically like, why hasn't anyone noticed? But then I thought, if they're N and S pole, well, maybe just no-one's noticed.

My second thought was, a good follow-up story would be an expedition to one of those points, and what you do see if you experiment with it?

I assumed the characters would not initially have the resources to go to one of those remote places, but there might be far-flung parallel worlds where scientists have expanded understand of them and governments trade across boundaries etc.

But I like your answer too, that it's plenty infinite, but only so much on any particular loop. Or maybe, they're all infinite, but some of them dead-end in uninteresting worlds without life.

I'm not sure I want everywhere to have a *fixed* number of points. If it's more than 2, I'm not sure I want to fix it. I'm not sure if that works mathematically, I guess, there's a point where if you keep travelling, the S pole discontinuity splits into two, initially too close together to traverse, then a tight figure-8, eventually it migrates to the bermuda triangle?