Cut out a disk from a stretchable cloth or plastic. Cut out a smaller
disk from the center of the disk. The surface you have formed is called
an annulus.
Questions:
How many boundaries did the disk have?
How many boundaries does the annulus have?
What is the Euler characteristic of the disk?
What is the Euler characteristic of the annulus?
What effect did cutting out a disk have on the Euler characteristic of
the disk?
Can the annulus be deformed into a tube?
What are the Euler characteristics of the annulus and the tube?
Are the annulus and tube topologically equivalent?
Is an annulus?
(Note: The white
triangle is a hole in the black surface.)
Activity 4:
Create a sphere by blowing up a round balloon and tying it shut. (Note:
Disregard the tubular extension as part of the spherical surface.) Tile
the sphere with colored markers.
Questions:
What is the sphere's Euler characteristic?
Is a sphere topologically equivalent to a tube?
Is a sphere topologically equivalent to a disk?
Is a sphere topologically equivalent to an annulus?
Is a sphere with a disk cut out topologically equivalent to a disk?
Is a sphere with two disks cut out topologically equivalent to a tube?
What effect would cutting a disk out have on the sphere's Euler
characteristic?
Is creating a hole in a surface the same as removing a disk?
Which of the following deformations change the Euler characteristic of a
surface:
an annulus?