YAN JIE (Ryan)
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’
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
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
Actually these two are also homeomorphic
We should know that in the topology, as long as we don’t the
original structure, any stretch and deformation is accepted.
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
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.
O An n-manifold is a topological space that
“locally looks like” the Euclidian space Rn
O Topological space: set properties
O Euclidian space: geometric/coordinates
O A sphere is a 2-manifold
O A circle is a 1-manifold
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.
O A surface in R3 is called orientable, if we
can clearly distinguish two
sides(inside/outside above/below)
O 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.
O Genus 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, sphere
O Genus 1: torus, coffee cup
O Genus 2: the symbols 8 and B
Euler characteristic function
If M has g holes and h boundary components then
= 2 – 2g – h
O (M) is independent of the polygonization
(M)
O Torus ( =0, g=1)
O double torus ( = -2 , g=2)
 =1
=2
=0
 = -2
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
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
Firstly we should change the map by replacing
areas of land by points and the by arcs.
The 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.
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 experiment
O 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 belt
OBecause of one side property, when we make the
Mobius strip-like conveyor belt, both sides of belt will
be used.
Another experiment
O 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 amazement
O 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?
Three dimensions
O 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 Bottle
O 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.