Tag Archives: algebraic topology

Enclosing closed curves in squares

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Let’s look at the following innocent looking question:

Is it possible to circumscribe a square about every closed curve?

The answer is YES! I found an unexpected and interesting proof in the book “Intuitive Combinatorial Topology ” by V.G. Boltyanskii and V.A. Efremovich . Let’s now look at the outline of proof for our claim:

1. Let any closed curve K be given. Draw any line l and the line l’ such that line l’ is parallel to l as shown in the fig 1.

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2. Move the lines l and l’ closer to K till they just touch the curve K as shown in fig 2. Let the new lines be line m and line m’. Call these lines as the support lines of curve K with respect to line l.

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3. Draw a line l* perpendicular to l and the line (l*)’ parallel to l* . Draw support lines with respect to line l* to the curve K as shown in the fig 3. Let the rectangle formed be ABCD .

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4. The rectangle corresponding to a line will become square when AB and AD are equal . Let the length of line parallel to l (which is AB)  be h_1(\mathbf{l}) and line perpendicular to l (which is AD) be h_2(\mathbf{l}). For a given line n, define a real valued function f(\mathbf{n}) = h_1(\mathbf{n})-h_2(\mathbf{n}) on the set of lines lying outside the curve .  Now rotate the line l in an anti-clockwise direction till l coincides with l’. The rectangle corresponding to l* will also be ABCD (same as that with respect to l). When l coincides with l’, we can say that  AB = h_2(\mathbf{l^*}) and AD = h_1(\mathbf{l^*}).

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5. We can see that when the line is lf(\mathbf{l}) = h_1(\mathbf{l})-h_2(\mathbf{l}). When we rotate l in an anti-clockwise direction the value of the function f changes continuously i.e. f is a continuous function (I do not know how to “prove” this is a continuous function but it’s intuitively clear to me; if you can have a proof please mention it in the comments). When l coincides with l’ the value of f(\mathbf{l^*}) = h_1(\mathbf{l^*})-h_2(\mathbf{l^*}). Since h_1(\mathbf{l^*}) = h_2(\mathbf{l}) and h_2(\mathbf{l^*}) = h_1(\mathbf{l}). Hence f(\mathbf{l^*}) = -(h_1(\mathbf{l}) - h_2(\mathbf{l})). So f is a continuous function which changes sign when line is moved from l to l’. Since f is a continuous function, using the generalization of intermediate value theorem we can show that there exists a line p between l and l* such that f(p) = 0 i.e. AB = AD.  So the rectangle corresponding to line p will be a square.

Hence every curve K can be circumscribed by a square.

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Borsuk-Ulam Theorem

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Yesterday, I was fortunate enough to attend a lecture delivered by Dr. Ritwik Mukherjee, one of my professors, to motivate the study of algebraic topology. Instead of using the “soft targets” like Möbius strip etc. he used the following profound theorem for motivation:

If f: S^n \to \mathbb{R}^n is continuous then there exists an x\in S^n such that:  f(-x)=f(x).

This is known as Borsuk-Ulam Theorem. To appreciate this theorem, one need to know a fundamental theorem about continuous functions known as Intermediate Value Theorem:

If a continuous function, f, with an interval, [a, b], as its domain, takes values f(a) and f(b) at each end of the interval, then it also takes any value between f(a) and f(b) at some point within the interval.

Here is a video by James Grime illustrating Borsuk-Ulam Theorem in 3D:

Though the implications of the theorem itself are beautiful, following corollary known as Ham sandwich theorem is even more interesting. Here is a video by Marc Chamberland explaining this theorem:

Also, yesterday Grant Sanderson uploaded a video exploring the relation of Borsuk-Ulam Theorem with a fair division problem known as Necklace splitting problem:

But, to my amazement, this theorem is related to one of the other most astonishing theorem of algebraic topology called Brouwer fixed-point theorem:

Every continuous function from a closed ball of a Euclidean space into itself has a fixed point.

Here is a video by Michael Stevens illustrating Brouwer fixed-point theorem in some interesting cases:

 

Now the applications of this theorem are numerous, and there is a book dedicated to this theorem: “Fixed Points” by Yu. A. Shashkin. But my favourite application of this fixed point theorem is to the board game called Hex, explained by Marc Chamberland here:

If you come across some other video/article discussing the coolness of “Borsuk-Ulam Theorem” please let me know.

Edit (18 May 2018): Proof of Brouwer’s Fixed Point Theorem by Tai-Danae Bradley:

 

What is Topology?

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A couple of years ago, I was introduced to topology via proof of Euler’s Polyhedron formula given in the book “What is Mathematics?” by Richard Courant and Herbert Robbins. Then I got attracted towards topology by reading the book “Euler’s gem – the polyhedron formula and the birth of topology” by David S. Richeson. But now after doing a semester course on “introduction to topology” I have realized that all this was a lie. These books were not presenting the real picture of subject, they were presenting just the motivational pictures. For example, this is my favourite video about introduction to topology by Tadashi Tokieda (though it doesn’t give the true picture):

Few months ago I read the book “The Poincaré Conjecture” by Donal O’Shea and it gave an honest picture of algebraic topology. But, then I realized that half of my textbook on topology is about point-set topology (while other half was about algebraic topology). This part of topology has no torus or Möbius strip (checkout this photo) but rather dry set theoretic arguments. So I decided to dig deeper into what really Topology is all about? Is is just a fancy graph theory (in 1736, both Topology and graph theory started from Euler’s Polyhedron formula) or it’s a new form of Geometry which we study using set theory, algebra and analysis.

The subject of topology itself consists of several different branches, such as:

  • Point-Set topology
  • Algebraic topology
  • Differential topology
  • Geometric topology

Point-set topology grew out of analysis, following Cauchy’s contribution to the foundations of analysis and in particular trigonometric representation of a function (Fourier series). In 1872, Georg Cantor desired a more solid foundation for standard operations (addition, etc.) performed on the real numbers. To this end, he defined a Cauchy sequence of rational numbers. He creates a bijection between the number line and the possible limits of sequence of rational numbers. He took the converse, that “the geometry of the straight line is complete,” as an axiom (note that thinking of points on the real line as limits of sequence of rational numbers is “for clarity” and not essential to what he is doing). Then Cantor proved following theorem:

If there is an equation of form \displaystyle{0=C_0+C_1+\ldots +C_n+\ldots} where C_0 = \frac{d_0}{2} and C_n = c_n\sin{(nx)} +d_n\cos{(nx)} for all values of x except those which correspond to points in the interval (0,2\pi) give a point set P of the \nuth kind, where \nu signifies any large number, then d_0=1, c_n=d_n=0

This theorem lead to definition of point set to be a finite or infinite set of points. This in turn lead to definition of cluster point, derived set, …. and whole of introductory course in topology. Modern mathematics tends to view the term “point-set” as synonymous with “open set.” Here I would like to quote James Munkres (from point-set topology part of my textbook):

A problem of fundamental importance in topology is to find conditions on a topological space that will guarantee that it is metrizable…. Although the metrization problem is an important problem in topology, the study of metric spaces as such does not properly belong to topology as much as it does to analysis.

Now, what is generally publicised to be “the topology” is actually the algebraic topology. This aspect of topology is indeed beautiful. It lead to concepts like fundamental groups which are inseparable from modern topology. In 1895, Henri Poincaré topologized Euler’s proof of Polyhedron formula leading to what we call today Euler’s Characteristic. This marked the beginning of what we today call algebraic topology.

For long time, differential geometry and algebraic topology remained the centre of attraction for geometers.But, in 1956, John Milnor discovered that there were distinct different differentiable structures (even I don’t know what it actually means!) on seven sphere. His arguments brought together topology and analysis in an unexpected way, and doing so initiated the field of differential topology.

Geometric topology has borrowed enormously from the rest of algebraic topology it has returned very scant interest on this “borrowed” capital. It is however full of problems with some of the simplest, in formulation, as yet unsolved. Knot Theory (or in general low-dimensional topology) is one of the most active area of research of this branch of topology. Here I would like to quote R.J. Daverman and R.B. Sher:

Geometric Topology focuses on matters arising in special spaces such as manifolds, simplicial complexes, and absolute neighbourhood retracts. Fundamental issues driving the subject involve the search for topological characterizations of the more important objects and for topological classification within key classes.
Some key contributions to this branch of topology came from Stephen Smale (1960s), William Thurston (1970s), Michael Freedman (1982), Simon Donaldson (1983), Lowell Edwin Jones (1993), F. Thomas Farrel (1993), … and the story continues.

References:

[1] Nicholas Scoville (Ursinus College), “Georg Cantor at the Dawn of Point-Set Topology,” Convergence (May 2012), doi:10.4169/loci003861

[2] André Weil, “Riemann, Betti and the Birth of Topology.” Archive for History of Exact Sciences 20, no. 2 (1979): 91–96. doi:10.1007/bf00327626.

[3] Johnson, Dale M. “The Problem of the Invariance of Dimension in the Growth of Modern Topology, Part I.” Archive for History of Exact Sciences 20, no. 2 (1979): 97–188. doi:10.1007/bf00327627.

[4] Johnson, Dale M. “The Problem of the Invariance of Dimension in the Growth of Modern Topology, Part II.” Archive for History of Exact Sciences 25, no. 2–3 (December 1981): 85–266. doi:10.1007/bf02116242.

[5] Lefschetz, Solomon. “The Early Development of Algebraic Topology.” Boletim Da Sociedade Brasileira de Matemática 1, no. 1 (January 1970): 1–48. doi:10.1007/bf02628194.