THE TOPOLOGICAL CLASSIFICATION OF SURFACES - TOPOLOGY - What Is Mathematics? An Elementary Approach to Ideas and Methods, 2nd Edition (1996)

What Is Mathematics? An Elementary Approach to Ideas and Methods, 2nd Edition (1996)

CHAPTER V. TOPOLOGY

§4. THE TOPOLOGICAL CLASSIFICATION OF SURFACES

1. The Genus of a Surface

Many simple but important topological facts arise in the study of two-dimensional surfaces. For example, let us compare the surface of a sphere with that of a torus. It is clear from Figure 135 that the two surfaces differ in a fundamental way: on the sphere, as in the plane, every simple closed curve such as C separates the surface into two parts. But on the torus there exist closed curves such as C’ that do not separate the surface into two parts. To say that C separates the sphere into two parts means that if the sphere is cut along C it will fall into two distinct and unconnected pieces, or, what amounts to the same thing, that we can find two points on the sphere such that any curve on the sphere which joins them must intersect C. On the other hand, if the torus is cut along the closed curve C’, the resulting surface still hangs together: any point of the surface can be joined to any other point by a curve that does not intersect C’. This difference between the sphere and the torus marks the two types of surfaces as topologically distinct, and shows that it is impossible to deform one into the other in a continuous way.

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Fig. 135. Cute on sphere and torus.

Next let us consider the surface with two holes shown in Figure 136. On this surface we can draw two non-intersecting closed curves A and B which do not separate the surface. The torus is always separated into two parts by any two such curves. On the other hand, three closed nonintersecting curves always separate the surface with two holes.

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Fig. 136. A surface of genus 2.

These facts suggest that we define the genus of a surface as the largest number of non-intersecting simple closed curves that can be drawn on the surface without separating it. The genus of the sphere is 0, that of the torus is 1, while that of the surface in Figure 136 is 2. A similar surface with p holes has the genus p. The genus is a topological property of a surface and remains the same if the surface is deformed. Conversely, it may be shown (we omit the proof) that if two closed surfaces have the same genus, then one may be deformed into the other, so that the genus p = 0, 1, 2, · · · of a closed surface characterizes it completely from the topological point of view. (We are assuming that the surfaces considered are ordinary “two-sided” closed surfaces. In Article 3 of this section we shall consider “one-sided” surfaces.) For example, the two-holed doughnut and the sphere with two “handles” of Figure 137 are both closed surfaces of genus 2, and it is clear that either of these surfaces may be continuously deformed into the other. Since the doughnut with p holes, or its equivalent, the sphere with p handles, is of genus p, we may take either of these surfaces as the topological representative of all closed surfaces of genus p.

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Fig. 137 Surfaces of genus 2.

*2. The Euler Characteristic of a Surface

Suppose that a closed surface S of genus p is divided into a number of regions by marking a number of vertices on S and joining them by curved arcs. We shall show that

(1)    VE + F = 2 – 2p,

where V = number of vertices, E = number of arcs, and F = number of regions. The number 2 – 2p is called the Euler characteristic of the surface. We have already seen that for the sphere, VE + F = 2, which agrees with (1), since p = 0 for the sphere.

To prove the general formula (1), we may assume that S is a sphere with p handles. For, as we have stated, any surface of genus p may be continuously deformed into such a surface, and during this deformation the numbers VE + F and 2 – 2p will not change. We shall choose the deformation so as to ensure that the closed curves A1, A2, B1, B2, · · · where the handles join the sphere consist of arcs of the given subdivision. (We refer to Fig. 138, which illustrates the proof for the case p = 2.)

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Fig. 138

Now let us cut the surface S along the curves A2, B2, · · · and straighten the handles out. Each handle will have a free edge bounded by a new curve A*, B*, · · · with the same number of vertices and arcs as A2, B2, · · · respectively. Hence VE + F will not change, since the additional vertices exactly counterbalance the additional arcs, while no new regions are created. Next, we deform the surface by flattening out the projecting handles, until the resulting surface is simply a sphere from which 2p regions have been removed. Since VE + F is known to equal 2 for any subdivision of the whole sphere, we have

VE + F = 2 – 2p

for the sphere with 2p regions removed, and hence for the original sphere with p handles, as was to be proved.

Figure 121 illustrates the application of formula (1) to a surface S consisting of flat polygons. This surface may be continuously deformed into a torus, so that the genus p is 1 and 2 – 2p = 2 – 2 = 0. As predicted by formula (1),

VE + F = 16 – 32 + 16 = 0.

Exercise: Subdivide the doughnut with two holes of Figure 137 into regions, and show that VE + F = –2.

3. One-Sided Surfaces

An ordinary surface has two sides. This applies both to closed surfaces like the sphere or the torus and to surfaces with boundary curves, such as the disk or a torus from which a piece has been removed. The two sides of such a surface could be painted with different colors to distinguish them. If the surface is closed, the two colors never meet. If the surface has boundary curves, the two colors meet only along these curves. A bug crawling along such a surface and prevented from crossing boundary curves, if any exist, would always remain on the same side.

Moebius made the surprising discovery that there are surfaces with only one side. The simplest such surface is the so-called Moebius strip, formed by taking a long rectangular strip of paper and pasting its two ends together after giving one a half-twist, as in Figure 139. A bug crawling along this surface, keeping always to the middle of the strip, will return to its original position upside down. The Moebius strip has only one edge, for its boundary consists of a single closed curve. The ordinary two-sided surface formed by pasting together the two ends of a rectangle without twisting has two distinct boundary curves. If the latter strip is cut along the center line it falls apart into two different strips of the same kind. But if the Moebius strip is cut along this line (shown in Figure 139) we find that it remains in one piece. It is rare for anyone not familiar with the Moebius strip to predict this behavior, so contrary to one’s intuition of what “should” occur. If the surface that results from cutting the Moebius strip along the middle is again cut along its middle, two separate but intertwined strips are formed.

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Fig. 139. Forming a Moebius strip.

It is fascinating to play with such strips by cutting them along lines parallel to a boundary curve and 1/2, 1/3, etc. of the distance across.

The boundary of a Moebius strip is an unknotted closed curve which can be deformed into a flat one e.g. a circle. During the deformation, the strip may be allowed to intersect itself so that a onesided selfintersecting surface results as in Figure 140 known as a cross-cap. The locus of selfintersection is regarded as two different lines, each belonging to one of the two portions of the surface which intersect there. The one-sidedness of the Moebius strip is preserved because this property is topological; a one-sided surface cannot be continuously deformed into a two-sided surface. Strikingly enough it is even possible to conduct the deformation in such a way that the boundary of the Moebius strip becomes flat, e.g. triangular, while the strip remains free from selfintersections. Figure 141 indicates such a model, due to Dr. B. Tuckermann; the boundary is a triangle defining one half of one diagonal square of a regular octahedron; the strip itself consists of six faces of the octahedron and four rectangular triangles, each one fourth of a diagonal plane.

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Fig. 140. Cross-cap

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Fig. 141. Moebius strip with plane boundary.

Another interesting one-sided surface is the “Klein bottle.” This surface is closed, but it has no inside or outside. It is topologically equivalent to a pair of cross-caps with their boundaries coinciding.

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Fig. 142. Klein bottle.

It may be shown that any closed, one-sided surface of genus p = 1, 2, · · · is topologically equivalent to a sphere from which p disks have been removed and replaced by cross-caps. From this it easily follows that the Euler characteristic VE + F of such a surface is related to p by the equation

VE + F = 2 – p.

The proof is analogous to that for two-sided surfaces. First we show that the Euler characteristic of a cross-cap or Moebius strip is 0. To do this we observe that, by cutting across a Moebius strip which has been subdivided into a number of regions, we obtain a rectangle that contains two more vertices, one more edge, and the same number of regions as the Moebius strip. For the rectangle, VE + F = 1, as we proved on page 239. Hence for the Moebius strip VE + F = 0. As an exercise, the reader may complete the proof.

It is considerably simpler to study the topological nature of surfaces such as these by means of plane polygons with certain pairs of edges conceptually identified (compare Chapt. IV, Appendix, Article 3). In the diagrams of Figure 143, parallel arrows are to be brought into coincidence—actual or conceptual—in position and direction.

This method of identification may also be used to define three-dimensional closed manifolds, analogous to the two-dimensional closed surfaces. For example, if we identify corresponding points of opposite

faces of a cube (Fig. 144), we obtain a closed, three-dimensional manifold called the three-dimensional torus. This manifold is topologically equivalent to the space between two concentric torus surfaces, one inside the other, in which corresponding points of the two torus surfaces are identified (Fig. 145). For the latter manifold is obtained from the cube if two pairs of conceptually identified faces are brought together.

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Fig. 143. Closed surfaces defined by coördination of edges in plane figure.

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Fig. 144. Three-dimensional torus defined by boundary identification.

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Fig. 145. Another representation of three-dimensional torus. (Figure cut to show identification.)