34
$\begingroup$

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)

$\endgroup$
9
  • 7
    $\begingroup$ Once the ant triangulates the surface, it can apply the standard algorithms for classification of surfaces to identify it uniquely as a sphere, connected sum of $g$ tori or connected sum of $k$ projective planes. Everything else will follow from that. $\endgroup$
    – Matt F.
    Sep 6, 2020 at 12:13
  • 3
    $\begingroup$ I would say that 2 is more equivalent to 3, since the hairy ball theorem reduces to the Euler characteristic (being the sum of the indices of a vector field). The results of 4 and 5 sort of involve Euler characteristic too, but are more subtle, so I would not call them equivalent. $\endgroup$ Sep 6, 2020 at 14:13
  • 2
    $\begingroup$ As Mike Shulman points out, 2 and 3 are closely related. Other than that relationship, I would say that all of the methods are distinct. $\endgroup$ Sep 6, 2020 at 16:40
  • 2
    $\begingroup$ To see the difference between 1 and 2, you could step up one dimension and note that $S^3$ is simply connected (i.e. has trivial fundamental group, every loop can be shrunk to a point) but does admit nowhere vanishing tangent vector fields (so its hair can be combed). $\endgroup$ Sep 6, 2020 at 18:05
  • 2
    $\begingroup$ Develop space travel, obviously. $\endgroup$
    – Asaf Karagila
    Sep 6, 2020 at 18:36

4 Answers 4

18
$\begingroup$

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:)

$\endgroup$
3
  • $\begingroup$ Mirco last time I asked about ants and you gave a great answer. This time I am asking about spiders ;-) mathoverflow.net/q/371606/156936 $\endgroup$
    – Claus
    Sep 13, 2020 at 17:38
  • $\begingroup$ Claus, thanks on 2 counts: for the compliment, and for the new challenge. Unfortunately, I am currently busy defending my last question from Asaf, and also I have to spend the weekend doing my own research ( I am a Data Scientist by trade, now). But, here is my promise: by next weekend, either someone has answered to your satisfaction, or I will done the spiderman suit and try my hand! :) viele Grüße, Mirco $\endgroup$ Sep 13, 2020 at 17:41
  • $\begingroup$ that's a great comment, thanks a lot!! In case you have time to look into it next weekend, please also tell me about this statement in your profile: "I do not believe in natural numbers, yet I reject no math". Very interested what it means. $\endgroup$
    – Claus
    Sep 13, 2020 at 17:53
11
$\begingroup$

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

$\endgroup$
2
  • 4
    $\begingroup$ I am very interested in your answer because you bring up persistent homology (and by the way your picture makes a great point!). Please let me ask two questions about your answer (1) if a distance function is required, but the ant in the OP cannot measure length - will distance function still work? (2) Does persistent homology also address the bullet points 3-5 in the OP, i.e. in the sense that it unifies all methods? $\endgroup$
    – MaryS.
    Sep 6, 2020 at 12:10
  • $\begingroup$ (1) if the ant cannot measure length, so if it has no notion of scale, then it has no way to exclude small perturbations of the surface (such as the arch) that increase its genus. (2) the approach gives the Betti numbers of the simplicial complex, so it fully characterizes the connectivity of the space; the coloring approach in point 5 of the OP identifies the chromatic number, which presumably is an independent characterization. $\endgroup$ Sep 6, 2020 at 12:23
2
$\begingroup$

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.

$\endgroup$
1
$\begingroup$

Without any metric: an ant living flat on a torus may exhibit a complete planar graph of five vertices ($K_5$). The it knows it does not live in a sphere. In fact the vertex chromatic number of the torus is 7. It works for n hole torus.

REMARK: It is not clear that living flat on the sphere you can show that $K_5$ is not planar. Meaning in general that differentiating a surface of type $A$ from a surface $B$ may not be symetrical.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy