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 deformation1、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

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

XY

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 PropertiesHomeomorphism 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 ManifoldsOAn n-manifold is a topological space

that “locally looks like” the Euclidian space Rn O Topological space: set propertiesO Euclidian space: geometric/coordinates

OA sphere is a 2-manifoldOA circle is a 1-manifold

Open vs. Closed SurfacesA 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.

OrientabilityOA surface in R3 is called

orientable, if we can clearly distinguish two sides(inside/outside above/below)

OA 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 holesOGenus of a surface is the

maximal number of nonintersecting simple closed curves that can be drawn on the surface without separating it

O Normally when we count the genus, we just count the number of holes or handles on the surface

O Example: O Genus 0: point, line, sphereO Genus 1: torus, coffee cupO Genus 2: the symbols 8 and B

Euler characteristic function

= 1 = 2 = 0

If M has g holes and h boundary components then (M) = 2 – 2g – h

O (M) is independent of the polygonizationO Torus ( =0, g=1)O double torus ( = -2 , g=2)

= -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 KonigsbergEuler’s theoremFour color problem

The Seven Bridges of Konigsberg

O 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.

O 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

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

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

O There are Three vertices with odd degree in the picture

O Take one of these vertices, we can see there are three lines connected to this vertex.

O There are two cases for this kind of vertices:O You could start at that vertex, and then arrive

and leave later. But then you can’t come back. O The first time you get to this vertex, you can

leave by another arc. But the next time you arrive you can’t.

O 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!!!!!!

O Thus it is impossible to draw the above picture in one pencil stroke without retracing.

O Thus we are unable to solve The Bridges of Konigsberg problem.

Möbius strip

How many sides has a piece of paper?O 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 experimentO Draw a line along the centre of your cylinder

parallel to one of its edges.

O Also do the same on your Möbius band

O What did you notice?

- A Möbius band has only one side.

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

Another experimentO What do you think would happen if you cut

along the line you’ve drawn on your cylinder?

O Will the same thing happen with the Möbius band?

O 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 amazementO Cut a new rectangle. You are going to draw two lines to

divide it into thirds.

O Now give it a twist and join the ends to form a Möbius band. Cut along one of the lines. What happens?

O You should get a long band and a short band.O Is the short band an ordinary band or a Möbius band?

Check by yourself after class

Three dimensionsO 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.

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

The Klein BottleO The result is a bottle with only one side, which we

should probably call the outside.

O It can’t be made; this is just an artist’s impression.

Only one surface!!

!

Thanks for listening!