2 Surface topology 2.1 Classification of surfaces In this second introductory chapter, we change direction completely. We dis- cuss the topological classification of surfaces, and outline one approach to a proof. Our treatment here is almost entirely informal; we do not even define precisely what we mean by a ‘surface’. (Definitions will be found in the following chapter.) However, with the aid of some more sophisticated technical language, it not too hard to turn our informal account into a precise proof. The reasons for including this material here are, first, that it gives a counterweight to the previous chapter: the two together illustrate two themes—complex analysis and topology—which run through the study of Riemann surfaces. And, second, that we are able to introduce some more advanced ideas that will be taken up later in the book. The statement of the classification of closed surfaces is probably well known to many readers. We write down two families of surfaces g, h for integers g ≥ 0, h ≥ 1. 2 2 The surface 0 is the 2-sphere S . The surface 1 is the 2-torus T .For g ≥ 2, we define the surface g by taking the ‘connected sum’ of g copies of the torus. In general, if X and Y are (connected) surfaces, the connected sum XY is a surface constructed as follows (Figure 2.1). We choose small discs DX in X and DY in Y and cut them out to get a pair of ‘surfaces-with- boundaries’, coresponding to the circle boundaries of DX and DY. Then we glue these boundary circles together to form XY. One can show that this resulting surface is (up to natural equivalence) independent of the choices of discs. Also, the operation is commutative and associative, up to natural equivalence. Now we define inductively, for g ≥ 2, 2 g = g−11 = g−1T (Figure 2.2), which we can write as 2 2 g = T ...T . Chapter2: Surface topology 11 X Y # X Y Figure 2.1 Connected sum g = 3 Figure 2.2 Genus-three surface There are many other representations of these surfaces, topologically equiv- alent. For example, we can think of g as being obtained by deleting 2g discs from the 2-sphere and adding g cylinders to form g ‘handles’. Or we can start with a disc and add g ribbons in the manner shown in Figure 2.3. The boundary of the resulting surface-with-boundary is a circle, and we form g by attaching a disc to this boundary to get a closed surface (i.e. a surface with no boundary). 2 The surface 1 is the real projective plane RP . We can form it by starting with a Möbius band and attaching a disc to the boundary circle. We cannot do this within ordinary three-dimensional space without introducing self- intersections: more formally, RP2 cannot be embedded in R3. But we can still perform the construction to make a topological space and, if we like, we can 12 Part I : Preliminaries Figure 2.3 Disc with handles think of embedding this in some Rn for larger n. Again, there are many other models possible. Notice that we can think of our Möbius band as a disc with a twisted ribbon attached (Figure 2.4). Then the construction falls into the same pattern as our third representation of g. Now we make the family of surfaces h by taking connected sums of copies 2 of 1 = RP : 2 2 h = RP ...RP . Now let S be any closed, connected, surface. (More precisely, we mean com- pact and without boundary, so, for example, R2 would not count as closed.) We say that S is orientable if it does not contain any Möbius band, and non- orientable if it does. This leads to the following statement. Classification Theorem. If S is orientable, it is equivalent to one (and precisely one) of the g. If S is not orientable, it is equivalent to one (and precisely one) of the h. (We emphasise again that this statement has quite a different status from the other ‘Theorems’ in this book, since we have not even defined the terms precisely.) To see an example of this, consider the ‘Klein bottle’ K. This can be pictured in R3 as shown in Figure 2.5, except that there is a circle of self-intersection. So, we should think of pushing the handle of the surface into a fourth dimension, where it passes through the ‘side’, just as we can take a curve in the plane as Chapter2: Surface topology 13 Figure 2.4 Disc with twisted ribbon—a Möbius band Figure 2.5 Klein bottle 14 Part I : Preliminaries Figure 2.6 Self-intersecting curve pictured in Figure 2.6 and remove the intersection point by lifting one branch into three dimensions. Now it is true, but perhaps not immediately obvious, that K is equivalent to 2. To see this, cut the picture in Figure 2.5 down the vertical plane of symme- try. Then you can see that K is formed by gluing two Möbius bands together along their boundaries. By the definition of the connected sum, this shows that K = RP2RP2, since the complement of a disc in RP2 is a Möbius band. Now we will outline a proof of the classification theorem. The proof uses ideas that (when developed in a rigorous way, of course) go under the name of ‘Morse theory’. A detailed technical account is given in the book by Hirsch (1994). The idea is that, given our closed surface S, we choose a typical real- valued function on S. Here ‘typical’ means, more precisely, that f is what is called a Morse function. What this requires is that if we introduce a choice of a gradient vector field of f on S, then there are only a finite number of points Pi , called critical points, in S where v (see below) vanishes, and near any of one of these points Pi we can parametrise the surface by two real numbers u, v such that the function is given by one of three local models: r u2 + v2 + constant; r u2 − v2 + constant; r −u2 − v2 + constant. The critical point Pi is said to have index 0, 1 or 2, respectively, in these three cases. For example, if S is a typical surface in R3, we can take the function f to be the restriction of the z co-ordinate (say): the ‘height’ function on S. The vector field v is the projection of the unit vertical vector to the tangent spaces of S, and the critical points are the points where the tangent space is horizontal. The points of index 0 and 2 are local minima and maxima respectively, and the critical points of index 1 are ‘saddle points’ (Figure 2.7). Chapter2: Surface topology 15 index 2 index 2 index 2 index 1 index 1 index 1 index 1 index 0 v index 1 f index 0 Figure 2.7 Indices of Morse function Now, for each t ∈ R, we can consider the subset St = {x ∈ S : f (x) ≤ t}. There are a finite number of special cases, the ‘critical values’ f (Pi ) where Pi is a critical point. If f is a sufficiently typical function, then the values f (Pi ) will all be different, so to each critical value we can associate just one critical point. If t is not a critical value, then St is a surface-with- boundary. Morover, if t varies over an interval not containing any critical values, then the surfaces- with-boundaries St are equivalent for different parameters t in this interval. To see this, if t1 < t2 and [t1, t2] does not contain any critical value, then one v deforms St2 into St1 by pushing down the gradient vector field (Figure 2.8). Now consider the exceptional case when t is a critical value, t0 say. The set St0 is no longer a surface. However, we can analyse the difference between St+ and St− for small positive . The crucial thing is that this analysis is concentrated around the corresponding critical point, and the change in the surface follows one of three standard local models, depending on the index. r { 2 v2 ≤±} Index 0. Near the critical point, St0± corresponds to u + , which is empty in one case and a disc in the other case. In other words, St0+ is obtained from St − by adding a disc as a new connected component. r 0 Index 2. This is the reverse of the index-0 case. The surface St0+ is formed by attaching a disc to a boundary component of St0−. r Index 1. This is a little more subtle. The local picture is to consider {u2 − v2 ≤±}, 16 Part I : Preliminaries t2 t1 St1 Figure 2.8 St as shown in Figure 2.9. This is equivalent to adding a strip to the boundary (Figure 2.10). Thus St0+ is formed by adding a strip to the boundary of St0−. We can now see a strategy to prove the Classification Theorem. What we should do is to prove a more general theorem, classifying surfaces-with- boundaries (not necessarily connected, and including the case of an empty boundary). Suppose we have any class of model surfaces-with-boundaries which is closed under the three operations associated to the critical points of index 0, 1 and 2 explained above. Then it follows that our original closed surface S must lie in this class, since it is obtained by a sequence of these operations.