What is topology? A branch of geometry Ignores differences in shapes caused by stretching and twisting without tearing or gluing. Math joke: – Q: What is a topologist? – A: Someone who cannot distinguish between a doughnut and a coffee cup.
Explanation of joke Michael Freedman, Fields Medal (1986) for his work in 4-dimensional topology ?=?=
Which surfaces look the same to a topologist? Note: no handles To a topologist, these objects are: torus Punctured torus sphere Punctured torus sphere
The punctured torus as viewed by various topologists
We can make all these shape ourselves!... topologically speaking What is this?
How do we make a two-holed torus? Hint: It’s two regular tori glued together. Find the gluing diagram
Pre-operative procedure: making a hole in the torus via its diagram
Making a two-holed torus out of 2 one-holed tori 1. Start with 2 one-holed tori: 2. Make holes in the diagrams. 3. Join holes. 3. Stretch it all out.
Note the pattern We can make a one-holed torus out of a rectangle. We can make a two-holed torus out of an octagon. Therefore, we can make an n-holed torus out of an 2n-gon. Ex: glue sides to get 6-holed torus We say this is a surface of genus n. n holes
What about an n-holed torus with a puncture???? Recall regular torus with holeNow fetch his orange brother Now glue them together Voila! A punctured two-holed torus What can you say about the blue/orange boundary?
Orientability Roughly this means that you can define an arrow pointing “OUT” or “IN” throughout the entire surface. Q: Are all tori orientable? A: Yes!
Is the Moebius strip orientable?
What can we glue to the boundary of the Moebius strip? Another Moebius strip to get a – Klein bottle A disk to get a – Projective plane Sliced up version
Are these surfaces orientable??
Classification of surfaces theorem Any non-infinite surface MUST be made up of a bunch of “bags” (both varieties may be used) and possibly a bunch of holes. For example: