NOTES FOR MATH 5520, SPRING 2011 DOMINGO TOLEDO 1. Outline This will be a course on the topology and geometry of surfaces. This is a continuation of Math 4510, and we will often refer to the notes for that course [12]. We first recall the definition of a surface. The first two parts are in Definition 6.1 of [12]. Definition 1.1. A topological space S is called: (1) A topological surface if it is a Hausdorff space with a countable basis and it has the property that every x 2 S has a neighborhood U 2 which is homeomorphic to an open set in R , in other words, there exists a covering fUαgα2A for some index set A, and for each α 2 A 2 there exists a homeomorphism φα : Uα ! Vα, where Vα ⊂ R is open. These homeomorphisms are called coordinate charts. (2) A differentiable surface or a smooth surface if it is a topological surface and the above homeomorphisms (or coordinate charts) can be chosen to have the following property: whenever Uα \Uβ 6= ;, the −1 homeomorphism φα ◦ φβ : φβ(Uα \ Uβ) ! φα(Uα \ Uβ) is smooth. −1 The maps φα ◦ φβ are called the transition maps between charts. (3) A topological surface with boundary is a Hausdorff space S with a countable basis with the property that every x 2 S has a neighbor- hood U and a homeomorphism φ : U ! V where V is an open set 2 2 in R+ = f(x; y) 2 R : y ≥ 0g. (4) If S is a topological surface with boundary and x and φ are as in the definition, the point x is called an interior point if φ(x) 2 f(x; y): y > 0g and X is called a boundary point if φ(x) 2 f(x; y): y = 0g. (5) A differentiable surface with boundary or smooth surface with bound- ary is a topological surface with boundary in which all the transition −1 maps φα ◦ φβ are smooth. Remark 1.1. (1) The corresponding definitions in any dimension n, with 2 n 2 n R replaced by R and R+ replaced by R+ = f(x1; : : : ; xn): xn ≥ 0g are called topological or differentiable n manifolds, or topological or differentiable n-manifolds with boundary. Date: April 29, 2011. 1 2 TOLEDO (2) It is true, but not trivial to prove, that the dimension n is a topo- logical invariant. It is also true but not trivial to prove that home- omorphisms take boundary points to boundary points and interior points to interior points. This last fact we will be able to prove for surfaces (n = 2) once we have the fundamental group. (3) The conditions of Hausdorff and countable basis are needed for cor- rectness of the definition, but it is not clear why they are required. One equivalent formulation of these conditions is to require that S be a metrizable space locally homeomorphic to the plane. I chose the formulation above because may be easier to check. We will not prove the equivalence of these two formulations, and for the most part we will forget these conditions since they will be automatic in the examples we consider. Example 1.1. Here are some examples of smooth surfaces. A good reference is the first chapter of [9]. In all these examples it will be clear that they n have a countable basis because R does, and all the examples are subspaces n or quotient spaces of subspaces of some R , so they still have a countable basis. The Hausdorff property would be easy to check in each case. 2 3 2 2 2 2 (1) The sphere S ⊂ R defined by S = f(x; y; z): x + y + z = 1g. It is checked in Example 6.3 of [12] that S2 is a smooth surface. (2) The torus T 2, that can be defined (up to homeomorphism, or up to diffeomorphism) in one of three equivalent ways: as S1 × S1, as 2 2 the identification space R =Z , or as the identification space [0; 1] × [0; 1]=((x; 0) ∼ (x; 1); (0; y) ∼ (1; y)). See Example 4.4 of [12] for more details. In particular, the neighborhoods pictured in Figures 2 4.2 and 4.3 of [12] are homeomorphic to disks in R , showing that T 2 is a topological surface, and a bit more care shows that it is a smooth surface. (3) The Klein Bottle K of Definition 4.5 of [12], see Figure 4.4. It is the identification space [0; 1] × [0; 1]= ∼, where the identification is (x; 0) ∼ (x; 1) and (0; y) ∼ (1; 1 − y). Again it is a smooth surface. (4) The M¨obiusband M of Definition 4.6 of [12]: M = [0; 1]×[−1; 1]=((0; y) ∼ (1; −y)). This is an example of a surface with boundary. 2 2 2 (5) The disk D = f(x; y) 2 R : x + y ≤ 1g and the cylinder C = [0; 1] × [−1; 1]=((0; y) ∼ (1; y)) are other examples of surfaces with boundary. (6) The projective plane P 2 is defined to be the quotient of S2 by the identification x ∼ −x. In other words, antipodal points on S2 are identified. Points in P 2 are in one to one correspondence with lines 3 2 through the origin in R , just as points in S are in one to one 3 correspondence with rays through the origin in R . To check that P 2 is a smooth manifold, let p : S2 ! P 2 be the projection map, that 5520 NOTES 3 2 2 ± to x 2 S assigns the antipodal pair fx; −xg 2 P . The six sets Uz , ± ± Uy , Ux , where the corresponding coordinate has the corresponding sign, used in Example 6.3 of [12] to cover S2 project to 3 sets that cover P 2. Namely since every antipodal pair has a representative (x; y; z) with one of x; y; z > 0, the ones with superscript + suffice. Call their images under p Uz;Uy;Ux, and use the same formulas for the transition maps to check that they are smooth. (7) Another way to describe the projective plane P 2 is as follows: Let 2 2 2 S+ = f(x; y; z) 2 S : z ≥ 0 and @S+ denote its boundary fz = 0g. 2 2 Consider the map f : S+ ! P obtained as the composition: 2 2 2 S+ ⊂ S ! P : 2 Then f is surjective, it is injective on the interior of S+ and 2 to one 2 2 on its boundary. It induces a continuous bijection g :(S+= ∼) ! P 2 where ∼ is the equivalence relation x ∼ −x for all x 2 @S+. It is easy to see that g is a homeomorphism (see Example 2.2 below), so we get that P 2 is homeomorphic to a hemisphere with antipodal points on its boundary identified. Since a hemisphere is homeomorphic to a disk, we get the final description: P 2 is homeomorphic to a disk with antipodal points on its boundary identified. (8) The surface of genus 2 is defined at the end of x4 of [12] as an identification space of an octagon, see Figure 4.6 and the pictures on pp. 300{301 of [6], to justify that the following identifications give the picture of a surface \of genus two" or \with two holes". b1 a1 a1 b1 a2 b2 a2 b2 Figure 1.1. Surface of Genus Two (9) Presenting a surface as a quotient of a polygon: Before we define the surface of genus g, we explain a convenient notation for describing the quotient space of a polygon by identifying the of edges of its boundary in pairs, using Figure 1.1 as an example. We choose a direction around the perimeter of the polygon, say clockwise, label 4 TOLEDO the edges to be identified with the same letter, draw similar arrows on each of the two edges that are identified indicating how they are identified: a monotone map identifying tail with tail. Move around the perimeter and write down the symbols for the edges, either a letter, if you travel in the same direction as the arrow, or its inverse, if you travel in the opposite direction. Thus the identification in Figure 1.1 could be written as −1 −1 −1 −1 a1b1a1 b1 a2b2a2 b2 ; while the identifications in the boundaries of the following four squares, running counterclockwise from the bottom, can be written as aba−1b−1 gives T = torus T 2 aba−1b gives K = Klein bottle abb−1a−1 gives S = sphere S2 abab gives P = projective plane P 2 Figure 1.2. Four Surfaces as quotients of the square (10) Checking that the quotient space is a surface requires finding for each point in the identification space a neighborhood homeomorphic to 2 an open set in R , say, a disk. Disk neighborhoods are immediate for any point in the interior of the polygon. For boundary points that are in the interior of an edge a set that projects to a disk in the quotient space is illustrated in the first picture of Figure 1.3. For the vertices we have to check how many distinct ones there are in the quotient space. In the case of the octagon for the genus two surface all the vertices go to a single point in the quotient space, and set that projects to a disk neighborhood in the quotient space is as in the second picture of Figure 1.3 Observe that the parts of the neighborhood of an interior point of an edge fit not only topologically, but also geometrically, into a 5520 NOTES 5 Figure 1.3.