Jean-Michel KantorInstitut Mahématique de JUSSIEU Case 247 - 4, place Jussieu - 75252 Paris Cedex FRANCE
Topology, as its name indicates, is a (mathematical) way of
conceiving of TOPOS : the place, the space, all space, and everything
included in it. Let us begin with the space of the Greeks : in
the very first reflections on space, and starting with the point
that is the atom, we can note a double point of view which we
will again encounter later on. In fact, in ancient Greek there
are two words which are used for describing a point :
And so the term " analysis situs " was coined and would remain in use until the twentieth century. Topology was born in 1735 ( even if the term would only be created as of 1863), when Euler, a Basles native who had moved to St. Petersburg, reported the following problem: In Koenisberg (today known as Kaliningrad), there is an island (A) surrounded by a river which is split into two branches. Fig. 1 And Euler was asked the following question : could a person possibly manage to cross one single time over each bridge? Opinion was divided at the time, and Euler gave the general solution -which is valid for any number of bridges and in any distribution of the branches. What is essential here is that he found the solution because he understood that the problem did not depend on the precise map of the city: he stated that it was not a problem of geometry : the distances, the lengths of the bridges for example, and the angles, do not come into play, and Euler establishes the new nature of the problem by using the term "geometry of position", an expression introduced for the first time by Leibnitz for "determining position and for seeking the properties which result from this position, without regards to the sizes themselves" [Euler 1741]. In other words, the quintessence of the problem resides in Figure 2. Fig. 2 This is the first graph and the first manifestation of topology. The problem is reduced to its essence- a geometrical structure is transformed into a more flexible structure, that of topology: If we replace each bank by a point or a cross (as symbol of the corresponding bank), and each bridge by a line which joins the associated banks, we then replace Figure 1 with Figure 2. The problem is then to draw a path with a pencil from a given summit -- let's say A corresponding to the bank A, and ending at a given summit and following once, and only once, each of the lines ; the problem has no solution. In effect, let's imagine that we are following -- as we should -- the path and reach, at a point we'll call B, which corresponds to one of the banks. It is then necessary to leave from there ! But then by coupling the paths of arrival and of departure, we observe that at each intermediary summmit, there must be an even number of bridges. As this is not the case in the map of Koenisberg, Euler's excursion is impossible. To summarize: the Koenisberg topology is the given in Figure 2. We can vary the geometric data (the length of the branches, the surface measurement of the islands ), deform the map of the city, but the topological structure and the resulting answer does not change. By Euler's promenade, mathematics suddenly discovers a brand new sense of freedom, which will then be constantly applied to forms and to spaces : now they can "twist again"! In this mathematical discipline, we no longer distinguish between two figures, two spaces, if you can pass from one to the other by means of a continuous deformation -- with neither leap nor cut. Topology is like mathematics made of rubber. Fig. 3 Figure 3 gives an example which we often summarize by affirming that topology is the field in which you no longer distinguish between the cup and the breakfast bun. The use of graphs such as the one in Figure 2 makes it possible to pose - with a diagram - the questions in which only the combinational is of importance, in spite of their complexity, for example all questions which arise from the organization of tasks, or networks (Figure 4). You can well imagine the influence technique can have on the theory of graphs! Fig. 4
The theme of knots has inspired an immense body of literature in so many fields : artistic [Coomaraswamy 1944], ethnographic, and scientific before becoming the object of an advanced mathematical theory [Belpoliti and Kantor 1996; Sossinski 1999]. We may borrow several illustrations from Albert Flocon, an artist and professor from the Bauhaus school, who has both written and drawn on the subject of knots with the imagination of an artist fascinated with topology (Fig. 5). We may well wonder what exactly is the cause of this universal interest. Fig. 5 It is true that knots are a simple way of escaping from the obtuseness of space : the presence of the knot overthrows the milieu itself. From the labyrinths of Leonardo to the medieval illuminations, and from Eskimo games to fishing techniques, the theme of the knot is widespread throughout all of the world's cultures. And this is an elementary example of topology: the deformations of the string are exactly the deformations which are authorized for the topologist. The mathematician wonders just how to recognize the way that a complex knot is, in fact, undone. Or again, he poses a simple mathematical problem which could easily have been one of Alice's games (the young friend of Lewis Carroll) when she went through the looking glass. Can you deform the clover knot (hitch) of Figure 6 A to change it into the clover knot of Figure 6 B which happens to be its very image in the looking glass? Fig. 6a (left) and 6b (right, mirror image) In the middle of the nineteenth century, physics attempted to understand electricity, and the environment in which it was propagated: was the space then filled? Did ether exist? A physicist named Thomson, the future Lord Kelvin, imagined any atom as a circular vortex in the midst of the ether. It then becomes crucial to classify these knots, and hopefully to recover the classification of elements of Mendeleieff. Although these reflections weren't successfully concluded in physics (even though you can perceive, in the past few years, an avatar in them in the theory of strings), mathematicians have taken over the question and have obtained, after lengthy efforts, the complete classification of knots.
Fig. 7 Mathematicians in the early nineteenth century then asked, "Why not try to apply to surfaces the same classifications used for knots?" They then began by creating a veritable bestiary of surfaces:
so goes a student song from Tom Lehrer of Cambridge, Massachusetts. Fig. 8. Klein's bottle Fig. 9 Fig. 10 Flocon even took it upon himself (for fun) to cut the Moebius strip in three (Fig. 9). Klein's bottle (Fig. 8) which fixes neither interior or exterior, evokes another refrain -- also from Tom Lehrer:
Classification required and was eventually accomplished by Riemann (1826-1866). The result was that surfaces can be classified according the number of holes along them (the type of surface), and that this number can be determined "without leaving the surface." In other words, a little mathematician ant moving across the surface of Figure 8 , for example, could determine the type of surface on which it lives, without leaving that surface.
On the other hand, for each of the preceeding stages -- point, knot, surfaces -- we considered figures which had a " degree of liberty ", zero for the point, where no movement is possible, one for the knot, and two for the surfaces. This number of degree of liberties is called dimension ; thus the torus (Figure 7) may then be represented by a couple of two numbers, each of which represents an angle : this is a variety of dimension 2. Inversely, thanks to Riemann we can see appearing spaces of dimension four and five, etc. As the space-time of Einstein's relativity brought into play a space of dimension 4, at the beginning of the century we were witness to a cultural fashion. Salvador Dali associates the body of Christ with the hypercube ("Corpus Hypercubus") and Max Weber depicted the "Interior of the fourth Dimension". Let us not forget, however, that this veritable mental liberation which opened the path to both modern topology and to a part of contemporary mathematics, has remained confidential : the common mental universe is still regulated and limited by the banal coordinates of Descartes. A few words are in order here to evoke the figure of Henri Poincaré - the founding father of modern topology ; and in particular for his famous conjecture which is possibly on the way to being solved [Perelman]: it consists in characterizing the sphere of dimension three (analogous to the sphere of Figure 7) by a property of loops analogous to those described in paragraph III.2. [2]
Space is our common ground, and it's impossible to break free of it, even in our dreams. The great Master of the universe is an architect, and ever since Galileo we have been imagining Him speaking and communicating in the language of mathematics. And since Riemann and Poincaré, He can dream of other spaces, and proceed to build them as a highly-gifted topologist. This discipline has far from exhausted its resources; Bachelard was already a portender of Lacan and his borromean knots:
More recently, the cognitive sciences have been trying to create topological models of brain functioning (an old tradition compared memory to a theater [Yates 2001]). Topology lends itself to comparisons and metaphors, for its flexibility is inscribed in its very structure : we can deform objects, as long as it's done with gentleness and subtlety! We have been able to get a feel for an underground path network of new ideas, all the way from Bauhaus -- here represented by Flocon -- to the situationist topology of Asger Jorn [1960]. Perhaps in the future it will be useful for the organization of space, by creating a Floconian bridge between the construction of individual space and the organization of social life?
Euclidis Prota]. return to
text[2] In more precise terms, Poincaré's conjecture
states that if a smooth variety of dimension three is such that
any closed loop can be continuously reduced to a point, then
it is equivalent to the sphere of dimension three.
Internationale situationniste 5.Bachelard, Gaston and Albert Flocon. 1950.
Ballestro, Catherine. 2000. Belpoliti, M. and J.M. Kantor. 1996. Coomaraswamy Ananda Kentish. 1944. Deleuze G. 1988. Euler, L. 1741. Solutio problematis ad geometriam
situs pertinentis, Flocon, A. 1975.
A.T. Fomenko. 1966. Leibniz, Lietzman, W. 1965. Le Lionnais, François , ed. 1997. Patras Frédéric. 1999. Géométries.
Topologies. In: Perelman, Grisha. Finite extinction time for the solutions to the Ricci flow on certain three-manifolds. http://arXiv.org/pdf/math.DG/0307245. (See also previous articles on Ricci curves.) Sossinski, Alexei. 1999. Yates F. 2001.
La
Quinzaine littéraire. For more, see http://www.math.jussieu.fr/~kantor.
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