An intelligent ant living on a torus or sphere – Does it have a universal way to find out?
Carlo's answer is definitely pointing in the right direction: simplicial complexes or more generally, simplicial sets, are conjured up by most points points mentioned by the PO (certainly 1 3, 4. 5 perhaps, with a twist, and as for 2, no idea) .
Unfortunately, as indicated by Carlo's comments, it falls short on one requirement, that ants do not know anything about metric spaces (nor what you can build on them, such as differential geometry).
Poor ants live in a world whose departments of math contain only three courses (*):
- finite combinatorics
- topology (presumably also finite)
- basic logic
Topological Data Analysis starts off with a cloud set of points immersed in a metric space (mostly euclidean $R^n$, but not necessarily).
Its main tool is persistent homology, which creates a filtration of simplicial complexes, thereby providing different views of $X$ at different resolution scales.
Where do these simplicial complexes come from ? They are Vietoris-Rips Complexes (see here; essentially you use the distance between groups of points to fill your simplexes).
So, no metric no Rips complex.
But (there is always a but in life): perhaps not all is lost.
What about creating a filtration of complexes by-passing entirely the metric?
Yes, sounds good, you may say, but how? Well, in ants world they have basic topology. So, for instance, suppose an ant goes from A to B, it can tell if during her trip she met point C (ie she can tell whether C is in some 'edge" between A and B). Similarly, given a set of distinguished points $A_0, \ldots A_n$ , she can tell whether they are indipendent, ie none of them lies in some slice of ant-world which is span by some subset. The independent subsets will become higher simplices (this approach is basically the one folks in matroid theory take)
Assuming this bare bone capability, Carlo's answer can indeed be vindicated: the ants build their filtration of complexes by selecting larger and larger finite subsets of their world.
Of course, unless their world is also finite, there is no guarantee that they will ever find out its final topology.
(*) on the funny side (apologies to serious MO fellows): trying to think of Ant's World I found out that is very much to my liking, especially the Departments of Math. A non Cantorian, non Dedekind paradise . Perhaps I should move there for a change:)
I presume the OP has in mind the topological distinction between a sphere and a torus, so the method should apply to deformed surfaces. A meaningful/universally valid method for this purpose must include the notion of "scale". Otherwise we would conclude that the earth is a torus, or even a surface of higher genus. Such a method is offered by the framework of persistent homology.
Space is represented by a simplicial complex with a distance function. Loops such as provided by the arch shown here can then be identified and excluded depending on the scale on which they occur. Efficient algorithms exist to identify the scale-dependent homology group, and thus obtain the topological invariants.
Aloba arch, Wikimedia Commons
If an ant has an ability to draw different colors of line, and detect when it has crossed a line, it could start by drawing a red/green line pair, with green on the right, and wander around until it hits the line. If it encounters the green side of the line and follows the line to the left, it would reach the start point and thus be able to close the loop. If it encounters the red side, following the line to the right would let it return to the start and close the loop. In either case, it could select two new colors, arbitrarily select the red or green side of the loop, and resume wandering. If it encounters a line of the new colors, it should close that loop using the same strategy for red/green. If it encounters a previously drawn loop, it may cross it, but should keep track of which side of each loop it is on.
No amount of wandering would allow the ant to prove it was on a sphere. If, however, it were to encounter a loop from one side having last left it via the other, that would prove that the loop which was thus encountered went around a hole, and thus that the structure had at least one.