Classification of Surfaces

Transcription

1 Classification of Surfaces 16 April 2014 Classification of Surfaces 16 April /29

2 Last Time On Monday we saw some examples of surfaces and how we can build some by starting with a rectangle and gluing edges. Classification of Surfaces 16 April /29

3 Building Surfaces From Little Panels We haven t, and won t, give a formal definition of surface. But, one way to think about them is that a surface, being a two-dimensional object, can be built by gluing together little square panels together. Classification of Surfaces 16 April /29

4 The Euler Characteristic of a Surface If you build a surface with little panels, then you have effectively drawn a map on that surface. We can then talk about the vertices, edges, and faces of the map. If V is the number of vertices, E the number of edges, and F the number of faces, we define the Euler characteristic of the surface as V E + F We saw last week Euler s formula, which says in this terminology, that the Euler characteristic of a sphere is 2. Essentially, Euler s formula is talking about a sphere. Classification of Surfaces 16 April /29

5 There can be many ways to build a surface from little panels. It is not at all obvious that one gets the same Euler characteristic, regardless of how the surface is built. However, it can be proven that this is the case. The Euler characteristic is a property of surfaces that does not change when you smoothly deform the surface. That is, if you reshape the surface with stretching or shrinking, the Euler characteristic does not change. Cutting and gluing can effect the Euler characteristic. Classification of Surfaces 16 April /29

6 The Euler Characteristic of a Sphere One way to see that the Euler characteristic of a sphere is 2 is to consider the following map There are two vertices and two edges (the forward half circle and the back half circle). There are two faces of the map. Therefore, V = E = F = 2, so V E + F = 2. Classification of Surfaces 16 April /29

7 We are using the fact that a hemisphere is topologically equivalent to a rectangle. That is, we can take a rectangle and smoothly deform it into a semicircle. Here is a video that shows this. Classification of Surfaces 16 April /29

8 Clicker Question What is the Euler characteristic of a Single Panel? To help answer this, what is the simplest map you can draw and cover the panel? How many vertices, edges, and faces does the map have? Classification of Surfaces 16 April /29

9 Answer A single panel has 1 face. It has 4 vertices (the four corners) and 4 edges. Therefore, the Euler characteristic is V E + F = = 1 Classification of Surfaces 16 April /29

10 The Euler Characteristic of the Torus To calculate this we need to break the torus up into panels. We showed a picture of this earlier, but it is made up of lots of panels and counting them will be difficult. Here is a way to build the torus with panels. Classification of Surfaces 16 April /29

11 Clicker Question How many faces are there? Answer: There are 32. There are 8 panels (4 blue, 4 yellow) in a ring around the hole. There are 4 such rings of panels, so 8 4 = 32 faces. Classification of Surfaces 16 April /29

12 Clicker Question How many vertices are there? Answer: There are 32. Looking at the circle on the top of the torus separating panels, we can count 8 vertices on the circle. There are four such circles, on the top, bottom, outside equator, and inside equator, so there are 8 4 = 32 total vertices. Classification of Surfaces 16 April /29

13 How many edges are there? There are 64. Looking at the four circles we mentioned in the previous slide, each is made up of 8 edges, so there are 8 4 = 32 of those. But there are also edges making up smaller circles in another direction. There are 8 smaller circles, each made up of 4 edges, so there are 32 of these. Together they give 64 edges. Classification of Surfaces 16 April /29

14 We then have V = F = 32 and E = 64. Then the Euler characteristic of the torus is V E + F = = 0 Thus, the Euler characteristic is 0. Classification of Surfaces 16 April /29

15 Orientability If you imagine walking on a surface, you may or may not be able to keep track of orientation. That is, you may not be able to keep track of up versus down, left versus right, etc. One way to think about this is to take a round trip path along the surface. When you finish, are you oriented in the same way as when you started? Perhaps the best example of a non-orientable surface is the Möbius strip. Classification of Surfaces 16 April /29

16 If you walk around the middle of the Möbius strip, after one complete circuit you ll end upside down. Classification of Surfaces 16 April /29

17 The Klein bottle is another example of a non-orientable surface. Classification of Surfaces 16 April /29

18 On the other hand, the sphere, torus, cylinder, and plane are all orientable. Classification of Surfaces 16 April /29

19 Boundary There is one other feature of surfaces, the boundary, that can help distinguish them. One way to think about the boundary is to break the surface into little panels. The boundary is then the collection of edges which are not glued to any other edge. For example, if our surface is a single panel, then the boundary consists of the four edges of the rectangle. Classification of Surfaces 16 April /29

20 The boundary of a disk is the circular border of the disk. The boundary of a cylinder with no top or bottom consists of the two circles at the top and bottom of the cylinder. Classification of Surfaces 16 April /29

21 One way to see that the border of a cylinder consists of these circles is to remember how we can construct a cylinder from a rectangle. We do this by gluing a pair of opposite sides, but we don t glue the other two sides. Those become the top and bottom circle. Classification of Surfaces 16 April /29

22 Some surfaces have no boundary. For example, a sphere and a torus each has no boundary. For both surfaces, each edge of a panel is glued to some other edge. Classification of Surfaces 16 April /29

23 Multi-Holed Tori One can have a torus-like surface but with 2 or more holes. Classification of Surfaces 16 April /29

24 Building New Surfaces From Old by Gluing A two-holed torus can be built by taking two tori and gluing them together. Likewise, a three-holed torus can be built by gluing three tori together. If you think of surfaces as built from small panels, to glue two surfaces together, we make a panel on each of the same size, put glue on each of the two panels, and then glue the two panels together. For example, if we glue two spheres, or two cubes, together. The result is again a cube (although stretched in one direction). We can glue projective planes together, or Klein bottles, or any combinations of surfaces. It appears that we can get some pretty wild surfaces this way. Classification of Surfaces 16 April /29

25 Summary of Some Surfaces Surface Orientable Boundary χ Panel yes yes 1 Sphere yes no 2 Cylinder yes yes 0 Möbius strip no yes 0 Torus yes no 0 Klein Bottle no no 0 Projective Plane no no 1 np no no 2 n n-holed Torus yes no 2 2n In the table, χ is the Euler characteristic and np refers to gluing n projective planes together. Classification of Surfaces 16 April /29

26 Classification of Surfaces It turns out that we have discussed enough properties of surfaces to determine all surfaces and distinguish surfaces, at least for surfaces that don t go on forever. A surface without boundary is topologically equivalent to either (1) a sphere, (2) n projective planes glued together for some n, or (3) n tori glued together for some n. The Klein bottle is such a surface. It turns out that it is equivalent to two projective planes glued together. Furthermore, two surfaces without boundary are equivalent if they have the same Euler characteristic and are both orientable or not-orientable. From the chart above, this gives the previous statement about the Klein bottle. Classification of Surfaces 16 April /29

27 Surfaces With Boundary Surfaces with are classified by their boundary, their orientability, and their Euler characteristic. That is, if two surfaces have the same type of boundary, the same Euler characteristic, and are both orientable or both non-orientable, then they are topologically equivalent. Classification of Surfaces 16 April /29

28 Next Week and Homework #8 Next week we will discuss interest rates. We ll see how savings accounts, loans, and annuities work. Homework #8 is on the website. It is due Friday 25 April. Classification of Surfaces 16 April /29