Doughnuts and Dumplings are Distinct: Homopoty-101

As everyone knows, a torus is different from a sphere. Topology is the study of properties that remain unchanged under continuous distortions. A square can be deformed into a circle or a sphere into an ellipsoid, whether flat like an orange or long like a lemon or banana.

Technically, sets are topologically equivalent if there is a homeomorphism between them, that is, a one-one map that is continuous in both directions — aka a bicontinuous bijection. The sphere ${\mathbb{S}^2}$ and torus ${\mathbb{T}^2 = \mathbb{S}^1\times\mathbb{S}^1}$ are topologically distinct: there is no homeomorphism between them. But how do we know this?

Experiments confirm that we cannot deform a balloon into an inner tube by stretching without tearing, but this is not a proof. You may search for a week, a year or a decade and you will not find a bicontinuous bijection. But, again, that proves nothing: absence of evidence is not the same as evidence of absence. We must prove that no homeomorphic mapping between ${\mathbb{S}^2}$ and ${\mathbb{T}^2}$ exists. This is not so easy.

The answer is a hole-in-one

The sphere and torus are both 2-manifolds and, locally, they are alike. But the torus has a “hole”. Can we use this to distinguish between the two sets? Yes!

If we draw a simple closed curve on the sphere, it can be continuously contracted to a point. For example, the Equator on the globe can be moved continuously northwards until it becomes a point at the North Pole. But on a torus, we can draw a loop around the ridge or through the hole. Neither of these can be contracted to a point while remaining on the surface of the torus.

The Fundamental Group

The equivalence between loops that can be continuously distorted into each other is called homotopy and we can treat all loops that are homotopic as being equivalent. We can combine two loops by concatenating them, or invert a loop by reversing its direction. In this way, homptopically distinct loops are found to form a group — the fundamental group. But this is an algebraic concept. We have just dipped our toes into the great ocean of algebraic topology.

Since all loops on the sphere are homotopic to each other, the fundamental group of the sphere is the trivial group ${\mathbb{I}}$ containing only the identity mapping. For the torus, with two distinct types of non-contractable loops — going through the hole or around the ridge — the fundamental group is ${\mathbb{Z}\times\mathbb{Z}}$ .

For each (path-connected) topological space ${X}$ , we can define a fundamental group ${\pi_1(X)}$ . It turns out to be a topological invariant: if two spaces ${X}$ and ${Y}$ are homeomorphic, then their fundamental groups are isomorphic: ${\pi_1(X) \simeq \pi_1(Y)}$ . But the fundamental groups of the sphere and torus are:

$\displaystyle \pi_1(\mathbb{S}^2) = \mathbb{I} \qquad\mbox{and}\qquad \pi_1(\mathbb{T}^2) = \mathbb{Z}\times\mathbb{Z} \,.$

Since these are distinct, the torus and sphere are not topologically equivalent. The doughnut and dumpling are distinct.

A Few Words on Algebraic Topology

We have seen how combining algebra and topology can be beneficial. Algebraic topology uses tools from abstract algebra to study topological spaces. The aim is to find algebraic invariants that classify these spaces up to homeomorphism. Homotopy groups are used in such classifications. Topological spaces with different homotopy groups are never homeomorphic, but topological spaces that are not homeomorphic can have the same homotopy groups. Thus, it is frequently the case that the classification is only up to homotopy equivalence, rather than homeomorphy.

The simplest homotopy group is the fundamental group. As we saw above, it holds information about the types of loop in a space. The fundamental group is only the first member of an infinite set of homotopy groups. And, for another day, there are cohomotopy groups, homology groups, cohomology groups and so on.

We can define a topological group to be one in which the group operations — multiplication and inversion — are continuous under the topology. However, we should point out that the fundamental group is not a topological group, as it has only algebraic struture; we have not defined a topology (a set of open sets) on ${\pi_1(X)}$ .

Sources

${\bullet}$ McCarty, George, 1967: Topology: an Introduction with Application to Topological Groups. McGraw-Hill Book Company. 270pp [see Chapter VIII: The Fundamental Group].

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Online Course

One advantage of the pandemic is that university courses that are normally delivered through lectures are now available throughout the world, online via Zoom sessions. A UCD Zoom course, on recreational mathematics, AweSums: Marvels and Mysteries of Mathematics, will be presented by Prof Peter Lynch, School of Mathematics & Statistics, UCD. Registration is now open here

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