Tin Can Topology
Frank Sottile
We will try to deconstruct and then construct the space of lines through the origin in R3.
That is, we first decompose this complicated set of lines into simpler sets, each of which we identify with an
ordinary object, and then we discuss how to glue these pieces together.
It is hopefully less complicated than the topology you would see in a topology class.
I usually do this in talks to students or classes that I teach.
It helps to make a Möbius band: Take a piece of paper, and cut it into (preferably three) strips of equal width lengthwise.
Tape them together edge-to-edge to make a long strip of paper.
Wait to make this into a Möbius band until paragraph 4 below.
Begin by imagining a can (more like a tuna can that is wide and not high) centered at the origin.
Each line through the origin hits the can twice, and there are two populations of lines.
Those with high slope that meet the can's top and bottom, and those with low slope that meet the can's sides.
(This second set forms the lines that are "near" the equator in the sphere.)
Those with high slope that meet the can's top and bottom–they are parametrized by
either the top or the bottom, which is a disc having as boundary a circle.
Pick the top.
Those with low slope meet the side of the can. Each meets the side in two points.
Cut the side vertically, say between the front and back halves.
Keep the back half.
Then almost all lines that meet the side meet this back half in a single point.
The exception are those that meet the two vertical edges between the front and back.
If you consider how the lines that meet the right vertical edge near its top
meet the left vertical edge near its bottom, you see that if you identify the two edges with a half-twist,
then you have a figure whose points correspond to the lines that meet the side of the can---this is
your Möbius band.
(If you haven't already, make it now.)
In the homework, the low slope lines meet the sphere in a neighbourhood of the equatorial circle.
Now that we have identified the high slope lines with the disc that is the top of the can, and the low slope lines with the Möbius band
formed from the side of the can, let us consider how to glue these together to form the real projective plane.
Note that the lines that are both low slope and high slope are those that meet the circles between the top/bottom and the side,
and this set of lines corresponds to one of those circles.
For the high slope lines, this is the circle that is the boundary edge of the top of the can.
For the low slope lines, this is the circle that forms the edge of the Möbius band.
We form the real porjective plane by glueing these two figures–the disc and the Möbius band–along this common edge.
Warning, this is impossible to do in R3 without a self-intersection in the surface.
For fun, look up pictures of the Boy surface (someone's name, a student of David Hilbert), especially the
sculpture of it
outside the Mathematisches Forschungsinstitut Oberwolfach.
Last modified: Subn Apr 14 21:36:38 PDT 2024