Whence topology: problems

I like to study topology. These first few blog posts will explain what it’s about. This first discusses some motivating problems.

The problem of the Königsberg bridges is widely regarded as one of the first in topology. Königsberg, then a city of Prussia, was cut into four regions, two sides and two islands, by the river Pregel.

The problem was to take one’s evening walk across each bridge exactly once. Euler’s insight into this problem was the right choice of abstraction. If any region has an odd number of bridges incident to it, there is no such evening walk possible, for any encounter with a region must involve walking on to it and off of it (or off and on). Neither the lengths of the bridges nor their number is important, but only the number of bridges incident to each region matters.

Perhaps a more substantive example would be that of integrating differential forms. As a simplest example, consider the case of line-integrals. Quoting from Maxwell’s Treatise on Electricity and Magnetism, Vol. 1, Art. 16,

[Consider] a point \(P\) on a line whose length, measured from a certain point \(A\), is \(s\)….

Let \(R\) be the … vector quantity at \(P\), and let the tangent to the curve at \(P\) make with the direction of \(R\) the angle \(\epsilon\), then \(R \cos \epsilon\) is the resolved part of \(R\) along the line, and the integral

\[L = \int_0^s R \cos \epsilon\,ds\]

is called the line-integral of \(R\) along the line \(s\).

This quantity is, in general, different for different lines drawn between \(A\) and \(P\). When, however, within a certain region, the quantity \([R \cos \epsilon\,ds]\) is an exact differential within that region, the value of \(L\) … is the same for any two forms of the path between \(A\) and \(P\), provided the one form can be changed into the other by continuous motion without passing out of this region. (Emphasis mine.)

In fact, to guarantee invariance of the integral, one can use either the exactness of the differential or the continuous deformation. In any event, Maxwell notes in the article immediately following that

We are here led to considerations belonging to the Geometry of Position, a subject which, though its importance was pointed out by Leibnitz and illustrated by Gauss, has been little studied.

This is, of course, a reference to topo-logy (study of place), which by now is quite well-studied.

As a yet more practical concern, consider the idea of dependence on initial conditions. Ideally, we would like a small change in the initial parameters of an experiment to effect only a small change in the outcome of the experiment. Otherwise, we could hardly consider the experiment replicable—random noise in the input could drastically change the outcome.¹ We might say the outcome should depend continuously on the input, or that outcome is a continuous function of the input. This too is a fundamental topological notion.

Yet another old topological problem is the classification of knots. Lord Kelvin theorized that atoms were knotted loops of “ether”. His colleague Tait took up the problem of classifying the simplest knots. A knot is simply a loop nicely embedded in space. Circular loops still count as knots, even though they are not knotted. We call them unknots. Exactly one of the following diagrams, due to Ochiai, can be continuously deformed into a circular loop, and therefore is also an unknot. The other is knotted.

My favorite motivation for topology, however, is Euler’s “theorem” on polyhedra. A regular polygon is determined, up to similarity, by how many sides it has. It is reasonable to ask if there is a similar classification for polyhedra. For instance, Euclid famously classified the regular polyhedra called the Platonic solids. What Euler noticed is that for pretty much any polyhedron he could think of—Platonic solids, Archimedean solids, you name it—the number \(S\) of vertices (anguli Solidi) plus the number \(H\) of faces (Hedrae) exceeds the number \(A\) of edges (Acies) by exactly two.

This comes from his letter to Goldbach of 14 Nov. 1750:

(6.) In omni solido hedris planis incluso aggregatum ex numero hedrarum et numero angulorum solidorum binario superat numerum acierum, seu est \(H + S = A + 2\)…

Euler brags a little but gets irritated he can’t prove it yet:

Es nimmt mich Wunder, dass diese allgemeinen proprietates in der Stereometrie noch von Niemand, so viel mir bekannt, sind angemerkt worden; doch viel mehr aber, dass die fürnehmsten davon als theor. 6 et theor. 11 so schwer zu beweisen sind, denn ich kann dieselben noch nicht so beweisen, dass ich damit zufrieden bin.

Euler did eventually publish a “proof” of this result. The ensuing history is instructive for mathematicians who haven’t made many serious mistakes or seen many made. This particular mistake led eventually to the crowning achievement of 19th-century topology, the classification of compact surfaces. Even Eulers’ mistakes were fruitful.

References

1. James Clerk Maxwell, A treatise on electricity and magnetism, Vol. 1, 3rd ed.

2. Leonhard Euler, Correspondence with Christian Goldbach, OO863, Euler to Goldbach, 14 Nov. 1750.

3. Mitsuyuki Ochiai, Nontrivial projections of the unknot, Astérisque, 1990, 192, 7–10.

Footnotes

1. So, not a replicable experiment, unless this sort of variability was itself the point one was trying to make with the experiment. ↩