Topology – Intelligent Ant on a Torus or Sphere

Question

I wanted to ask a question about topological invariants and whether they are connected in a fundamental or universal way. I am not an expert in topology, so please let me ask this question by way of a simple example.

Imagine an intelligent ant living on a torus or sphere, and it wants to find out. Let’s further assume the ant does not have the capabilities to do geometrical measurements, i.e. it cannot measure length, angle, curvature, whether a line is straight, and so on. The only capabilities it has are topological, combinatorial, logical.
Now there are several ways that it can distinguish a sphere from the torus, like for example

1. Work with loops and determine the fundamental group.
2. “Comb” the surface (apply the Hairy Ball Theorem).
3. Triangulate the surface, count vertices, edges, and faces, and determine its Euler characteristic.
4. Draw the complete graph with five vertices $K_{5}$. If it can be drawn without any edges crossing, then it must be the torus.
5. Triangulate the surface and color the vertices. Minimize the number of colors, but make sure adjacent vertices have different colors. If more than four colors are needed, it must be the torus.

I am not an expert in this field, but I think No. 1 and 2 are fundamentally equivalent (applying the same fundamental topological concepts). I imagine that No. 3 and 4 are also fundamentally equivalent. I am not sure about No. 5, I think its relation to 3 and 4 is through Hadwiger’s conjecture.

My question, can it be shown that all these methods are, in some way, fundamentally resting on the same, deeper concept? Asking differently, is there an abstract, universal method from which all the other examples follow or can be derived?

I would be interested to learn whether category theory or homotopy type theory provide such a foundational, universal view on this classification problem. My dream answer would be if someone said something like “all your methods are examples of the universal property of …”, but maybe that’s expecting too much.

I would be grateful for any hint or reference. Thank you in advance!

EDIT: Just adding references to make the post more self-contained
Fundamental Group https://en.wikipedia.org/wiki/Fundamental_group
Hairy Ball or Poincaré Brouwer Theorem https://en.wikipedia.org/wiki/Hairy_ball_theorem
Euler Characteristic https://en.wikipedia.org/wiki/Euler_characteristic
Graphs on Torus https://en.wikipedia.org/wiki/Toroidal_graph
Four Color Theorem and generalization to torus https://en.wikipedia.org/wiki/Four_color_theorem
Hadwiger Conjecture https://en.wikipedia.org/wiki/Hadwiger_conjecture_(graph_theory)

Answer 1

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 (*):

1. finite combinatorics
2. topology (presumably also finite)
3. 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:)