Topology of Surfaces

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1 EM225 Topology of Surfaces Geometry and Topology In Euclidean geometry, the allowed transformations are the so-called rigid motions which allow no distortion of the plane (or 3-space in 3 dimensional geometry). The transformations are translations, rotations and reflections. Euclidean geometry transformations preserve distances and angles. Topological transformations by contrast are a much larger class of transformations which allow distortions of the space. They include: stretching bending A topological transformation is any continuous transformation (which is continuously invertible). Not allowed is tearing or cutting: Note that topological transformations do not preserve distances or angles. distances angles Also available at: Page 1

2 What is a surface? A surface (without boundary) is a space which is locally like a (possibly distorted) piece of the plane. More precisely, each point P on the surface has a neighbourhood that looks like a distorted flat unit disc D = {(x, y) 4 2 : x 2 + y 2 < 1} in the plane with the origin (0, 0) of D corresponding to P. D P For surfaces with boundary, some points have neighbourhoods that are like a distorted flat unit half disc. P Two surfaces are topologically equivalent (or homeomorphic) if one can be obtained from the other by a topological transformation; that is, a sequence of stretching and bending distortions. Example: a sphere is topologically equivalent to a cube. Imagine a (near) spherical balloon trapped inside a cubical box. Inflate the balloon what happens? EM225 Topology of Surfaces Page 2

3 Examples of surfaces Cylinder Möbius band Torus Klein bottle Note that the Klein bottle contains a subset that is a Möbius band. This is illustrated in the diagram below. EM225 Topology of Surfaces Page 3

4 Projective plane This is the topologists version of the projective plane the (two-dimensional) projective geometry that we looked at earlier. To see this, recall that a projective Point is a line through the origin in 4 3. Projective Point Any projective Point intersects the unit sphere in two antipodal points. Horizontal projective Points intersect the sphere in two equatorial points; all other projective Points intersect the sphere in a point in the northern hemisphere and a point in the southern hemisphere. Projective Point Projective Point EM225 Topology of Surfaces Page 4

5 If we consider only the southern hemisphere, then each projective Point is represented by a unique point on the southern hemisphere, except for horizontal projective Points which are represented by a pair of equatorial antipodal points. Therefore we can represent the projective plane as the southern hemisphere with equatorial points identified (or glued ) in pairs. Since a hemisphere is topologically equivalent to a disc, this is essentially the description of the projective plane given above. Note that the projective plane contains a Möbius band as a subset, illustratred in the diagram below. The Connected Sum Construction Let S 1 and S 2 be two surfaces. Their connected sum S 1 # S 2 is defined as follows. 1. Remove a small disc from each surface. S 1 S 2 EM225 Topology of Surfaces Page 5

6 2. Pull out the boundaries slightly. S 1 S 2 3. Glue the boundary circles together. S 1 # S 2 Connected sum of S with a torus. S T S # T S with handle added EM225 Topology of Surfaces Page 6

7 Connected sum of S with a projective plane First note that a projective plane with a hole is a Möbius band. The following diagrams illustrate the fact that a projective plane with a Möbius band removed is a disc. Hence it follows that a projective plane with a disc removed is a Möbius band. Hence the connected sum of a surface S with a projective plane is S with a hole cut out and a Möbius band sewn into its place. S P S # P Orientability A surface is non-orientable if and only if it contains a subset that is topologically equivalent to a Möbius band; otherwise it is orientable. Examples 1. The sphere, cylinder, torus, double torus etc are all orientable. 2. The Möbius band itself, the projective plane and the Klein bottle are all non-orientable. EM225 Topology of Surfaces Page 7

8 3. Are the following surfaces orientable or non-orientable? b (i) (ii) a c b a c c b c b a a The Classification Theorem of Closed Surfaces A classification theorem for a certain class of objects describes all objects in the class; it provides a list or lists (possibly infinite) of all the objects in the class. Classification theorems are amongst the most satisfying in mathematics; unfortunately they are quite rare. 1 Classification theorems answer the question what examples are there of these kinds of objects? in the most complete way possible. A classification theorem essentially says two things these (the objects in the list) are all examples of a particular class of objects and there are no others. The classification theorem for closed surfaces says exactly what are all (compact) surfaces without boundary. Compact here just means does not extend to infinity thus the plane 4 2 is not compact. There are two lists one for orientable surfaces and one for nonorientable surfaces. The classification theorem says that all surfaces (without boundary) can be built from just three basic surfaces (sphere, torus and projective plane) using the connected sum construction. Classification Theorem of Closed Surfaces Let S, T and P denote the sphere, torus and projective plane respectively. Any closed surface without boundary is topologically equivalent to exactly one of the following. 1. (Orientable) A sphere with n handles: S # nt = S # T # T #...# T for some n 0 n copies 2. (Non-orientable) A sphere with m Möbius bands sewn in: S # mp = S # P # P#... # P for some m > 0. m copies 1 One wonderful example of a classification theorem is the Classification Theorem for Finite Simple Groups. This theorem is the combined work of many mathematicians around the world. The proof took some 30 years to obtain and, when completed in about 1981, the proof ran to at least pages spread through many journal articles. This monumental achievement is the longest proof (so far!) in the history of mathematics. (See Chapter 1 of R Garnier & J Taylor, 100% Mathematical Proof, for further details.) EM225 Topology of Surfaces Page 8

9 Surfaces with boundary The theorem extends quite simply to surfaces with boundary: any surface with boundary is topologically equivalent to one of those above with k ( 0) holes (discs cut out). Euler characteristic Consider any plane drawing of a graph. The drawing divides the plane into a number of faces. a b f g h c e d Let v, e, f denote the numbers of vertices, edges and faces respectively. For the graph above v = 8, e = 10, f = 4 (if we count the infinite face). Calculate v e + f for several examples. What do you notice? (Note that each face must be topologically equivalent to a disc.) 2 Provided we include the infinite face, calculating v e + f for a graph drawn in the plane is equivalent to calculating v e + f for a graph drawn on the surface of a sphere. To see this, consider a sphere sitting on a plane with the south pole of the sphere S as the point of contact. Then projecting from the north pole N sends each point P on the sphere (except the north pole itself) to a unique point P on the plane. This mapping from the sphere (with north pole removed) to the plane is called stereographic projection. N P S P 2 Chapter 1 of Imre Lakatos, Proofs and Refutations: The Logic of Mathematical Discovery, gives a superb insight into the development of the Euler characteristic in the form of a classical Platonic dialogue between a teacher and several pupils. Well worth reading! EM225 Topology of Surfaces Page 9

10 Now do the same thing for the surface of a torus calculate v e + f for each of several graphs drawn on the surface of a torus. Example Euler characteristic Let S be a surface with a graph drawn on its surface such that each face is topologically equivalent to a disc. Then the number v e + f depends only on the surface and not on the graph. In other words, for any particular surface, v e + f always gives the same number. The number v e + f is called the Euler characteristic of the surface, denoted χ. We know: for the sphere, χ = 2 for the torus, χ = 0. Exercise/Investigation Determine the Euler characteristic of several (orientable and non-orientable) surfaces. Hence formulate conjectures for the Euler characteristics of the standard surfaces χ( S # nt) = χ( S # T # T #... # T) n copies copies χ( S # mp) = χ( S # P # P#...# P) m EM225 Topology of Surfaces Page 10

Lecture 7: Jan 31, Some definitions related to Simplical Complex. 7.2 Topological Equivalence and Homeomorphism

Lecture 7: Jan 31, Some definitions related to Simplical Complex. 7.2 Topological Equivalence and Homeomorphism CS 6170 Computational Topology: Topological Data Analysis University of Utah Spring 2017 School of Computing Lecture 7: Jan 31, 2017 Lecturer: Prof. Bei Wang Scribe: Avani Sharma,