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

- Work with loops and determine the fundamental group.
- “Comb” the surface (apply the Hairy Ball Theorem).
- Triangulate the surface, count vertices, edges, and faces, and determine its Euler characteristic.
- Draw the complete graph with five vertices $K_{5}$. If it can be drawn without any edges crossing, then it must be the torus.
- 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 methodfrom 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)

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