Invitation to Combinatorial Topology
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Invitation to Combinatorial Topology
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| ISBN-13: | 9780486147888 |
|---|---|
| Publisher: | Dover Publications |
| Publication date: | 08/13/2012 |
| Series: | Dover Books on Mathematics |
| Sold by: | Barnes & Noble |
| Format: | eBook |
| Pages: | 136 |
| File size: | 3 MB |
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Invitation to Combinatorial Topology
By Maurice Fréchet, Ky Fan, Howard W. Eves
Dover Publications, Inc.
Copyright © 1967 Prindle, Weber & Schmidt, IncorporatedAll rights reserved.
ISBN: 978-0-486-14788-8
CHAPTER 1
Topological Generalities
1. QUALITATIVE GEOMETRIC PROPERTIES
The properties of figures explicitly stated in elementary geometry are, for the most part, metric properties—that is to say, properties depending on size or measure. Among such properties are, for example, the congruence of two triangles, the equality of two angles, the condition for a quadrilateral to be a square, etc. But certain properties of figures are quite independent of size or measure and are not separately encountered in elementary geometry. On the one hand, for example, consider the interior of a circle, the interior of an ellipse, and the interior of a square, and on the other hand, a ring-shaped region formed by two concentric circles in the plane [a]. All these figures clearly have different metric properties. Nevertheless, intuition tells us that there are some properties which are common to the first three figures but which the last does not possess. For example, the first three figures have this property in common: the region bounded by any simple closed polygonal line lying in the interior of the figure also belongs entirely to the interior of the figure. It is clear that the circular ring does not have this property. One can thus say that there are certain qualitative properties that the interior of a circle, the interior of an ellipse, and the interior of a square possess in common, and which the circular ring does not possess.
Consider, further, a circular circumference in the plane. This circumference divides the rest of the plane into two parts, such that two points in a common part are always able to be joined by a polygonal line in the plane which does not cut the circumference, whereas every polygonal line in the plane joining two points belonging to the two different parts cuts the circumference. But one can suitably modify the metric nature of the circumference without altering this property: if we replace the circumference by an ellipse or by a simple closed planar polygonal line, this property still holds. In a general way, all these figures are particular cases of what is called a closed Jordan curve (see further, page 13). It is the celebrated Jordan theorem that states that every closed Jordan curve in a plane divides the rest of the plane into two parts (in the precise sense described above). We thus have a qualitative property of the plane. In spite of the intuitive evidence of Jordan's theorem, a proof of the theorem is not easy [b]. (There are more than a score of proofs of this theorem. For example, see C. JORDAN [12], pages 92–100; A. DENJOY [5]; DE KERÉKJÁRTÓ [14], page 21, and [15]; E. SCHMIDT [21]; F. SEVERI [23], pages 24–35.)
In order better to reveal the existence of qualitative properties of figures, we shall look at some problems which are, moreover, interesting in themselves.
2. COLORING GEOGRAPHICAL MAPS5
Cayley in 1878 attributed to De Morgan the announcement of the four-color theorem [c]. According to this theorem, if an arbitrary geographical map represents countries (or political subdivisions) by their boundaries [d], then no more than four colors are sufficient to color the map so that any two countries touching each other along a common boundary have different colors. (Moreover, Frederick Guthrie asserted in 1880 that his brother Francis Guthrie had given a proof of this theorem as early as thirty years before.) For some time various different but false proofs were held to be sound. Actually, no one as yet has proved this theorem. But it has been shown that five colors always suffice [e], and it is easy to see that three colors are not in general sufficient. Consider, for example, a map of seven regions A, B, C, D, E, F, G situated as shown in Fig. 1. One easily ascertains that four colors are needed to color these seven regions. [It is interesting to note that, on the map of France, the département of Seine-et-Oise and the neighboring départements (Seine, Oise, Seine-et-Marne, Loiret, Eure-et-Loire, Eure) have a mutual relationship of the type pictured in Fig. 1.] One can even construct a map of only four regions for which fewer than four colors will not be sufficient. Fig. 2 shows a map of four regions in which every two are adjacent; it therefore requires four colors to color the regions.
In generalizing the four-color problem, one can pose the following: Given a surface, determine the minimum number of colors sufficient to color all possible maps on this surface so that any two regions touching along a curve have different colors (but two regions having only a finite number of points in common may have the same color). This minimum number is called the chromatic number of the surface.
In the case of a plane or a sphere, the chromatic number appears in all likelihood to be 4. But this is a conjecture that is very difficult to establish [f]. Nevertheless, the problem has been resolved for several kinds of surfaces.
Take, for example, the torus, which is the surface obtained by rotating a circle about a nonintersecting line in its plane (Fig. 3). It has been shown that the chromatic number of the torus is 7. We shall content ourselves here with showing that fewer than seven colors are not sufficient. To accomplish this, it suffices to prove that it is possible to construct seven regions on the torus such that each pair of these regions touch each other along a curve. Now this is possible. Cut the torus along a generating circle, which will convert the surface into a deformed cylinder. Straighten this and cut the cylinder along one of its altitudes; a rectangle will be obtained (Fig. 4). It follows that the regions on the torus can be represented on the rectangle, care being taken to identify the parallel sides of the rectangle. In Fig. 5, a division of the torus into seven regions is represented on a rectangle, the four regions G of the rectangle forming only one region on the torus. Each pair of these seven regions touch each other along a curve [g].
Note that an analogous problem cannot be posed concerning spatial domains.
It was Frederick Guthrie who first made the observation that, for any positive integer n, one can construct n domains in space such that each pair of them touch each other along a surface, from which one concludes that one needs n colors to color the domains. A simple example is obtained in the following way. Cut (Fig. 6) a parallelepiped by a plane parallel to the base, obtaining in this way two superposed parallelepipeds. Divide these two parallelepipeds respectively into n partial parallelepipeds, numbered from 1 to n, one by planes parallel to one direction of the lateral faces and the other by planes parallel to the other direction of the lateral faces. Consider as domain number k (k = 1, 2, ..., n) the pair of partial parallelepipeds numbered k. It is seen that each pair of these n domains touch each other along a surface; in order to color them, then, one must have n colors.
3. THE PROBLEM OF NEIGHBORING REGIONS
This problem is intimately related to the problem of coloring geographic maps. The problem is to determine, for a given surface, the maximum number of regions [h] which can be constructed on the surface such that each pair of these regions touch each other along a curve. It is seen that the chromatic number of a surface is at least equal to this maximum number of regions adjacent in pairs, for to color the regions adjacent in pairs it is necessary to have as many colors as there are regions. We have seen (Fig. 2) that the maximum number of regions adjacent in pairs for the plane is at least 4. It can be shown [i] that it is exactly 4. In the case of the torus, the two numbers are equal; the common value is 7.
Let us mention the further problem of determining the maximum number of points on a given surface that can be joined in pairs by curves drawn on the surface with no two of the curves intersecting one another. Consider, for example, five points on a plane. There are ten pairs of these points. It is impossible to draw on the plane ten nonintersecting curves such that each pair of the five points constitutes extremities of one of the curves. In other words, if one succeeds in drawing nine curves, not intersecting one another, joining nine pairs of these points, every curve joining the tenth pair must cut at least one of the other nine curves (Fig. 7). The sought maximum number of points for the plane is thus less than 5. In fact, it is 4 [i]. And, similarly, one can also prove (see HILBERT and COHN-VOSSEN [11], page 295) that, for every surface, the maximum number in question is equal to the maximum number of regions adjacent in pairs [i]. In particular, there then exist on the torus seven points (but there do not exist eight) which can be joined in pairs by curves drawn on the torus with no two of these curves intersecting each other.
4. TOPOLOGY, INDIA-RUBBER GEOMETRY
In the preceding problems, one deals with neither size nor measure; one is concerned only with certain qualitative properties. And, moreover, these qualitative properties have a common character that is important to clarify.
Imagine a torus made of a kind of elastic matter like India-rubber, which we are able to distort as we wish in a continuous fashion without tearing. When the torus is deformed, without tearing or overlapping, into another surface, the answer to any one of the preceding problems for the resulting surface will be the same as for the torus. One can, for example, find seven regions adjacent in pairs on the torus; it is clear that the same can be said for the resulting surface.
Similarly, imagine a figure made of India-rubber and deform it without tearing or overlapping. Certain properties of the figure are changed by the deformation, but there are other properties that remain unchanged. Those properties that remain invariant under the deformation are the so-called topological properties of the figure. It is topological properties that one studies in topology.
For example, if the surface of a circle is made of India-rubber, one can deform it, without tearing or overlapping, into the surface of an ellipse or the surface of a square, etc. The properties common to the surface of a circle, to the surface of an ellipse, to the surface of a square, and, more generally, to all figures obtained by deforming the surface of a circle without tearing or overlapping, are studied in topology.
Thus, vaguely and roughly speaking, we can say that topology is the geometry of India-rubber figures. The concept of homeomorphism will allow us to make this precise.
5. HOMEOMORPHISM
When one distorts, without tearing or overlapping, a figure made of India-rubber, one notices that there are two remarkable relationships between the initial figure and that obtained by the deformation: to each point of the one figure corresponds one and only one point of the other, and to two neighboring points of the one correspond two neighboring points of the other. These two relationships can be expressed in a mathematical form that we are going to make precise, and they are, in fact, the only relationships that matter, as far as we are concerned, in a "deformation, without tearing or overlapping, of an India-rubber figure."
Let E and F be two figures (or two sets of points), distinct or not. Suppose that to each point of E there corresponds a definite point of F, the same point of F perhaps corresponding to several points of E, and, moreover, each point of F corresponding to at least one point of E. We shall say of each such correspondence that it determines a(unique) transformation f of the figure E onto the figure F. If b denotes the point of F corresponding to a point a of E, we write b = f(a), and say that b is the image, or the transform, of the point a, or that f transforms à into b.
Now let f be a transformation of E onto F. In general, to two distinct points of E may correspond the same point of F. In the particular case where to different points a1and a2 of E there always correspond two different points f (a1) and f (a2) of F, we say that f is a biunique transformation, or a biunique correspondence [j], between the points of E and those of F. There exists, then, for each point b of F, one and only one point a = g (b) of E such that f (a) = b, and the correspondence g is called the inverse transformation of f, a particular transformation of set F onto set E. The transformation g, which is the inverse of f, is often designated by f-1.
A transformation f of E onto F is continuous when points of E which are close to a point a of E are transformed into points of F which are close to the point f (a). More precisely, we say that f is continuous at a point a of E if, for each number η 0 there exists a number ε 0 such that, for every point x of E at a distance less than η from a, the point f (x) in F is at a distance less than ε from f (a). If f is continuous at each point of E, we say f is a continuous transformation of E onto F.
A biunique transformation f of E onto F is said to be bicontinuous if it is continuous from E onto F and if, moreover, the inverse transformation f is continuous from F onto E.
A biunique and bicontinuous transformation is called a homeomorphism or a topological transformation. If we allow ourselves to speak intuitively, we can say that a homeomorphism between two figures (or two sets of points) is a correspondence such that to each point of either one of the two figures corresponds one and only one point of the other and to two neighboring points of either correspond two neighboring points of the other.
We spoke of the "deformation, without tearing or overlapping, of an India-rubber figure" in order to introduce the idea of homeomorphism. Actually, the "deformation without tearing or overlapping" is only a very particular process for the concrete realization of a homeomorphism. It is possible to achieve a homeomorphism by other processes. For example, let us cut a torus along the circumference of a generating circle to obtain a surface in the form of a tube with two ends. From this surface we make a knot (Fig. 8), and then rejoin the two ends in such a manner that points which coincided on the torus coincide on the final surface. One then sees that there exists a biunique and bicontinuous correspondence between the torus and the surface of the knot. We have passed from the first surface to the second by means of cutting, deforming, and pasting. This very simple example shows that we cannot consider a homeomorphism as the exact mathematical translation of "deformation without tearing or overlapping," to which another mathematical concept, that of isotopy, better applies, and of which we shall say more later (page 16). We used "deformation without tearing or overlapping" only to emphasize the principal property of a homeomorphism—that of being a biunique and bicontinuous transformation [k].
Two figures (or two sets of points) are said to be homeomorphic if one can pass from one to the other by a homeomorphism, that is, if there exists a biunique and bicontinuous transformation between them. If E and F are homeomorphic, and similarly F and G, it is clear that E and G are homeomorphic.
Examples of homeomorphic figures are numerous. Two figures that are superposable by displacement are homeomorphic. It follows that to show that two figures E and G are homeomorphic, it suffices to prove that E is homeomorphic to a figure F superposable on G by displacement.
As we have just seen, a torus and the surface of a knot (Fig. 8) are two homeomorphic figures. The surface of a sphere S and that of a tetrahedron T are homeomorphic : by replacing, if necessary, the sphere S by an equal sphere S', we can suppose that the center of sphere S' is in the interior of the tetrahedron; we can then project the one surface onto the other from the center of the sphere. One easily sees that this projection establishes a homeomorphism between the two surfaces S' and T, whence, according to the preceding, S and T are also homeomorphic.
(Continues...)
Excerpted from Invitation to Combinatorial Topology by Maurice Fréchet, Ky Fan, Howard W. Eves. Copyright © 1967 Prindle, Weber & Schmidt, Incorporated. Excerpted by permission of Dover Publications, Inc..
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Table of Contents
ForewordTranslator's Preface
CHAPTER ONE. TOPOLOGICAL GENERALITIES
1. Qualitative Geometric Properties
2. Coloring Geographical Maps
3. The Problem of Neighboring Regions
4. "Topology, India-Rubber Geometry"
5. Homeomorphism
6. "Topology, Continuous Geometry"
7. "Comparison of Elementary Geometry, Projective Geometry, and Topology"
8. Relative Topological Properties
9. Set Topology and Combinatorial Topology
10. The Development of Topology
CHAPTER TWO. TOPOLOGICAL NOTIONS ABOUT SURFACES
11. Descartes' Theorem
12. An Application of Descartes' Theorem
13. Characteristic of a Surface
14. Unilateral Surfaces
15. Orientability and Nonorientability
16. Topological Polygons
17. Construction of Closed Orientable Surfaces from Polygons by Identifying Their Sides
18. Construction of Closed Nonorientable Surfaces from Polygons by Identifying Their Sides
19. Topological Definition of a Closed Surface
CHAPTER THREE. TOPOLOGICAL CLASSIFICATION OF CLOSED SURFACES
20. The Principle Problem in the Topology of Surfaces
21. Planar Polygonal Schema and Symbolic Representation of a Polyhedron
22. Elementary Operations
23. Use of Normal Forms of Polyhedra
24. Reduction to Normal Form: I
25. Reduction to Normal Form: II
26. Characteristic and Orientability
27. The Principle Theorem of the Topology of Closed Surfaces
28. Application to the Geometric Theory of Functions
29. Genus and Connection Number of Closed Orientable Surfaces
Bibliography
TRANSLATOR'S NOTES
Index