Topology YAN JIE (Ryan)

What is topology? Topology is the study of properties of a shape that do not change under deformation A simple way to describe topology is as a ‘rubber sheet geometry’

The rule of deformation 1、we suppose A is the set of elements before deformation, B is the set of elements after deformation. So set A is bijective to set B. (1-1 correspondence) 2、bicontinuous, (continuous in both ways) 3、Can’t tear, join or poke/seal holes

Example X Y A is homeomorphic to B A very simple example is blowing a balloon. As the balloon gets larger, although the shape and pattern of the balloon will change(such like sphere becomes oval and length, area and collinearity will change), there is still one correspondence on the pattern between balloon and inflated balloon(the adjacent point near point A is still adjacent to point A after inflation.) A is homeomorphic to B

Example Actually these two are also homeomorphic

Here are the deformation We should know that in the topology, as long as we don’t the original structure, any stretch and deformation is accepted.

Topological Properties Homeomorphism has several types we should determine: 1、Surface is open or closed 2、Surface is orientable or not 3、Genus (number of holes) 4、Boundary components

Surfaces Surface is a space which “locally looks like” a plane: --For example, this blue sphere is a earth, earth is so large that when we just locally choose a piece of land, it will look like flat and it is 2D surface.

Surfaces and Manifolds An n-manifold is a topological space that “locally looks like” the Euclidian space Rn Topological space: set properties Euclidian space: geometric/coordinates A sphere is a 2-manifold A circle is a 1-manifold

Open vs. Closed Surfaces A closed surface is one that doesn't have a boundary, or end, such as a sphere, or cube, or pyramid, cone, anything like that. The surface is closed if it has a definite inside and outside, and there is no way to get from the inside to the outside without passing through the surface. An open surface is a surface with a boundary, such as a disk or bowl that you can get to the end of.

Orientability A surface in R3 is called orientable, if we can clearly distinguish two sides(inside/outside above/below) A non-orientable surface can take the traveler back to the original point wherever he starts from any point on that surface. Actually this is called mobius strip, I will talk about later.

Genus and holes Genus of a surface is the maximal number of nonintersecting simple closed curves that can be drawn on the surface without separating it Normally when we count the genus, we just count the number of holes or handles on the surface Example: Genus 0: point, line, sphere Genus 1: torus, coffee cup Genus 2: the symbols 8 and B

Euler characteristic function If M has g holes and h boundary components then (M) = 2 – 2g – h (M) is independent of the polygonization Torus ( =0, g=1) double torus ( = -2 , g=2)  = 2  = 0  = 1  = -2

Early development of topology There have been some contents of topology in the early 18th century. People found some isolated problems and later these problems had significant effect on the formation of topology. The Seven Bridges of Konigsberg Euler’s theorem Four color problem

The Seven Bridges of Konigsberg In Konigsberg, Germany, a river ran through the city such that in its centre was an island, and after passing the island, the river broke into two parts. Seven bridges were built so that people of the city could get from one part to another.

The people wondered whether or not one could walk around the city in a way that would involve crossing each bridge exactly once.

So this question can be summarized as: 1、go through the 7 bridges once 2、no repetition

solution Firstly we should change the map by replacing areas of land by points and the by arcs.

solution The problem now becomes one of drawing all this picture without second draw.

There are Three vertices with odd degree in the picture Take one of these vertices, we can see there are three lines connected to this vertex. There are two cases for this kind of vertices: You could start at that vertex, and then arrive and leave later. But then you can’t come back. The first time you get to this vertex, you can leave by another arc. But the next time you arrive you can’t.

Thus every vertex with an ODD number of arcs attached to it has to be either at the beginning or the end of your pencil-path. The maximum number of odd degree vertices is 2!!!!!! Thus it is impossible to draw the above picture in one pencil stroke without retracing. Thus we are unable to solve The Bridges of Konigsberg problem.

Möbius strip

How many sides has a piece of paper? A piece of paper has two sides. If I make it into a cylinder, it still has two sides, an inside and an outside.

How many sides has this shape? Now we cut a rectangle 2 cm wide, but give it a twist before we join the ends. Möbius band is made!

An experiment Draw a line along the centre of your cylinder parallel to one of its edges. Also do the same on your Möbius band

What did you notice? - A Möbius band has only one side.

Möbius bands are useful! You should have found your band only had one edge. This has been put to lots of uses. One use is in conveyor belt Because of one side property, when we make the Mobius strip-like conveyor belt, both sides of belt will be used.

Another experiment What do you think would happen if you cut along the line you’ve drawn on your cylinder? Will the same thing happen with the Möbius band? Try it!

The result is: 1、for the normal cylinder, after cutting through, it will split into two ordinary band. 2、for Möbius strip, it will produce a larger band with double length of original length. Here we should know that that larger band is not Möbius strip.

More amazement Cut a new rectangle. You are going to draw two lines to divide it into thirds. Now give it a twist and join the ends to form a Möbius band. Cut along one of the lines. What happens?

Check by yourself after class You should get a long band and a short band. Is the short band an ordinary band or a Möbius band? Check by yourself after class

Three dimensions Up till now we have just looked at 2D shapes. And when we twist them, we need our three dimensional world. Mathematicians have wondered what would happen if they took a 3D tube and twisted it in a fourth dimension before joining the ends.

Unfortunately we can’t do that experiment in our world, but mathematicians know what the result would be.

Only one surface!!! The Klein Bottle The result is a bottle with only one side, which we should probably call the outside. It can’t be made; this is just an artist’s impression. Only one surface!!!

Thanks for listening!