Topology - PowerPoint Presentation

Slide1

Topology

YAN JIE

(Ryan)Slide2

What is topology?

Topology

is

t

he

study of properties of a shape that do not change under

deformation

A

simple way to describe topology is as a

‘

rubber sheet geometry

’Slide3

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 holesSlide4

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

X

YSlide5

Example

Actually these two are also

homeomorphic

Slide6

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.Slide7

Topological Properties

H

omeomorphism

has several types we should determine:

1

、

Surface

is open or closed2、Surface is orientable or not3、

Genus (number of holes)4、Boundary componentsSlide8

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.Slide9

Surfaces and Manifolds

An

n-manifold

is a topological space that

“

locally looks like

”

the Euclidian space 

R

n

Topological space:

set properties

Euclidian space: geometric/coordinates

A sphere is a 2-manifold

A circle is a 1-manifoldSlide10

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. Slide11

Orientability

A surface in

R

3

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.Slide12

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 Slide13

Euler characteristic function



=

1



=

2



=

0

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

)



=

-2Slide14

Early development of topology

There have been some contents of topology in the early 18

th

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 problemSlide15

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.Slide16

The people wondered whether or not one could walk around the city in a way that would involve crossing each bridge exactly once.Slide17

So this question can be summarized as:

1

、

go through the 7 bridges once

2

、

no repetitionSlide18

solution

Firstly we should change the map by

replacing areas of land by points

and

the

by

arcs.Slide19

solution

The problem now becomes one of drawing

all this

picture

without second draw.Slide20

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. Slide21

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.Slide22

Möbius stripSlide23

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.Slide24

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!Slide25

An experiment

Draw a line along the centre of your

cylinder

parallel to one of its edges.

Also

do the same on your

Möbius

bandSlide26

What did you notice?

-

A

Möbius

band has only one side.Slide27

M

öbius

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.Slide28

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!Slide29

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.Slide30

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?Slide31

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 classSlide32

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.Slide33

Unfortunately we can

’

t do that experiment in our world, but mathematicians know what the result would be.Slide34

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!!!Slide35

Tha

nks for listening!