Maurice Frechet’s invitation to combinatorial topology…hark all high schoolers :-)

Reference: Invitation to Combinatorial Topology by Maurice Frechet and Ky Fan, Dover Publications, Inc.

Foreword:

This book, may perchance, fall into the hands of a reader curious about mathematics, but who, before beginning to read our work, wishes to have an idea, expressed in nontechnical language and in a few lines, as to “what is topology (also called analysis situs). Henri Poincare, the founder of topology has had already given satisfactory answer to the reader in an intuitive and particularly striking way:

“Geometers usually distinguish two kinds of geometry, the first of which they qualify as metric and the second as projective. Metric geometry is based on the notion of distance, two figures are there regarded as equivalent when they are ‘congruent’ in the sense that mathematicians give to this word. (like in our Indian maths text books, SSS Test, SAS Test, RHS test…). Projective geometry is based on the notion of a straight line; in order for two figures there to be considered equivalent, it is not necessary that they be congruent; it suffices that one can pass from one to other by projective transformation, that is, that one can be the perspective of the other. This second body of study has often been called qualitative geometry, and in fact, it is if one opposes it to the first; it is clear that measure and quantity ( that is, length, areas, volumes, measures of angles, length of arcs…) play a less important role. This is not entirely so, however, The fact that a line is straight is not purely qualitative; one cannot assure himself that a line is straight without making measurements, or without sliding on this line an instrument called a straightedge, which is a kind of instrument of measure.

“But it is a third geometry from which quantity is completely excluded and which is purely qualitative; this is “analysis situs.” In this figure, two figures are equivalent whenever one can pass from one to the other by a continuous deformation, whatever else the law of this deformation may be, it must be continuous. Thus, a circle is equivalent to an ellipse or even to an arbitrary closed curve, but it is not equivalent to a straight line segment since this segment is not closed. A sphere is equivalent to any convex surface (NB here this word convex has same meaning as in a convex polygon); it is not equivalent to a torus since there is a hole in a torus (or say a donut ! 🙂 and in a sphere there is no hole. Imagine an arbitrary design and a copy of this same design executed by an unskilled draftsman; the properties are altered, the straight line drawn by an inexperienced hand have sufferend unfortunate deviations and contain awkward bends. From the point of view metric geometry, and even of projective geometry, the two figures are not equivalent; on the contary from the point of view of analysis situs, they are equivalent.

Analysis situs is a very important science for the geometer; it leads to a sequence of theorems as well as those of Euclid; and it is not this set of propositions that Riemann has built one of the most remarkable and most abstract theories of pure analysis. I will cite two of these theorems in order to clarify their nature: (1) Two plane closed curves cut each other in an even number of points (2) if a polyhedron is convex, that is, if one cannot draw a closed curve on its surface without dividing it into two, the number of edges is equal to that of the vertices plus that of the faces diminished by two., and this remains true when the faces and edges of the polyhedron are curved.

And here is what makes this analysis situs interesting to us; it is that geometric intution really intervenes there. When, in a theorem of metric geometry, one appeals to this intuition, it is because it is impossible to study the metric properties of a figure as abstractions of its qualitative properties, that is, of those which are the proper business of analysis situs. It has often been said that geometry is the art of reasoning correctly from badly drawn figures. This is not a capricious statement; it is a truth that merits reflection. But what is a badly drawn figure ? It is what might be executed by the unskilled draftsman spoken of earlier; he alters the properties more or less grossly; his straight lines have disquieting zigzags; his circles show awkward bumps. But this does not matter; this will by no means bother the geometer; this will not prevent him from reasoning correctly.

“But the inexperienced artist must not represent a closed curve as an open curve, three lines which intersect in a common point by three lines which have no common point, or a pierced surface by a surface without a hole. Because then one would no longer be able to avail himself of his figure and reasoning would become impossible. Intuition will not be impeded by the flaws of drawing which concern only metric or projective geometry; it becomes impossible as soon as these flaws relate to analysis situs.

“This very simple observation shows us the real role of geometric intuition; it is to assist this intuition that the geometer needs to draw figures, or at the very least to represent them mentally. Now, if he ignores the metric or projective properties of these figures, if he adheres only to their purely qualitative properties, it will be then and only then that geometric intuition really intervenes. Not that I wish to say that metric geometry rests on pure logic and that true intuition never intervenes there, but these are intuitions of a different kind, analogous to those which play the essential role in arithmetic and in algebra.”

Cheers,

Nalin Pithwa

Significance of Topology : Efremovitch et al

Reference: Intuitive Combinational Topology by V. G. Boltyanskii and V. A. Efremovich and translated by Abe Shenitzer, Universitext, Springer Publications.

The elementary ideas of topology are based on direct observation of the world around us. It is clear that the geometric properties of a figure are not exhausted by the metric properties (such as lengths, angles, and so on); there are things outside the bounds of traditional geometry. Thus, a curve (a rope, a wire, a long molecule) cannot be described by its length alone. Indeed, it can be closed or not, and if closed it can be knotted in complicated ways. Two or more closed curves can be linked in a variety of ways. Solids and their surfaces can have holes. What characterizes such properties of solids that they are unaltered by deformations resulting from arbitrary distortions that do not involve tearing. Such properties are called topological. In addition to elementary geometric figures, many purely mathematical objects have topological properties, and it is these that determines their importance.

It is easier to determine the existence of topological properties of figures than it is to create a “calculus” of such properties, that is, to develop a branch of mathematics with exact concepts, rigorous rules and methods, as well as mathematical formulas for the representation of topological magnitudes.

The earliest important insights and exact topological relations are due to Euler, Gauss and Riemann. But it is no exaggeration to say that topology as an independent discipline was created at the end of the nineteenth century by Henri Poincare. The evolution of topology and the solution of its intrinsic problems turned out to be difficult and prolonged; in fact, it extended over seventy to eighty years. Many deep discoveries were made, which led in a number of cases to a revision of its foundations. Some of the greatest mathematicians, including Russians, took part in this process of development. In the 1920’s, P S Urysohn and P S Alexandroff established in Moscow the Soviet school of topology. Until the end of the 1950s, mathematicians in other areas regarded topology as a beautiful but useless plaything. I freely admit that as a student in the 1950’s, I chose topology as my future area of research because I was captivated by its beauty and “otherness” (compared with the traditional areas of mathematics), and that for a long time, until the late 1960’s, I was dissatisfied with the nature of its development, marked as it was by a paucity of applications. Nonetheless, it is important to note that many beautiful topological results had by then obtained in areas such as function theory and complex analysis, qualitative theory of dynamical systems and partial differential equations, operator calculus, and even in algebra.

However, it was not until the early 1970’s that topological methods began to penetrate strongly the apparatus of modern physics. Today their importance for different areas of physics is beyond doubt. In particular, topological methods are used in field theory and general relativity, in the physics of the anisotropy of solid media, in the physics of low temperatures, and in modern quantum theory. This makes it necessary to publish sufficiently elementary popular books on topology and its applications, accessible (at least in part) to high school seniors and beginning undergraduates interested in the natural sciences and technology.

(the above are the words of S. P. Novikov, Editor of the Russian Original)

The following is based on the “Introduction by the Authors”

Topology is a young and very important branch of mathematics. The famous German mathematician Hermann Weyl said that “the angel of topology and devil of abstract algebra fight for the soul of each individual mathematical domain.” He thus pointed out the remarkable subtlety and beauty of topology as well as the extent to which all of modern mathematics is interlaced in a remarkable way with the ideas of topology and algebra. In recent years, topology has penetrated more and more into physics, chemistry and biology. The reader will find an example of the use of topological ideas in physics in V. P. Mineev’s supplement. However, it is difficult to enter the magical world of topology. Just as the scaffolding surrounding and unfinished building prevents one from perceiving the beauty of its design, so too the many tiresome details of the theory that fill the books on topology prevent one from seeing with the mind’s eye this beautiful mathematical structure. Even professional mathematicians often given up rather than face the difficulties barring the way to the mastery of topology (especially, algebraic topology).

All this makes it imperative to write popular books on topology. A first book of this kind was written in our country in the 1930’s (Paul Alexandroff and V A Efremovich, Short Survey of the Fundamental Ideas of Topology, Moscow, 1936). Then, beginning in 1957, our book Short Survey of the Fundamental Concepts of Topology appeared in installments in issues 2, 3, 4 and 6 of the Soviet journal Mathematical Education (translations of the book were published in Poland, Japan, and Hungary). Both of these books became bibliographical rarities long ago. Part of the material from our Short Survey is contained in the present book and represents V. A. Efremovich’s contribution (he was the moving spirit behind the creation of the Short Survey and the popularization of topology). The major part of the text is new and was written by me (V. G. Boltyanskii). This gave me the opportunity to include a few recent results. I have also added more than 200 problems, for I think that the study of a scientific, or popular-scientific, book is useful only if the reader reflects on the issues it deals with.

(By V. G. Boltyanskii)

Cheers,

Nalin Pithwa