Math 444/539 : Geometric Topology

Fall 2010

Syllabus

Lecture 1 : [pdf]
Intro to class, quotient topology.

Homework 1 : [pdf]

Lecture 2 : [pdf]
CW complexes

Lecture 3 : [pdf]
Manifolds, statement of classification of surfaces

Homework 2 : [pdf]

Lecture 4 : [pdf]
Triangulations, Euler characteristic, orientability
For a proof that every surface can be triangulated, there are several options. I first learned the proof from Chapter 1 of Ahlfors-Sario's book "Riemann Surfaces". The proof there is similar to Rado's proof. Another account at a similar level is in Moise's book "Geometric Topology in Dimensions 2 and 3". I recently had pointed out to me a more accessible proof in the paper "The Jordan-Schoenflies Theorem and the Classification of Surfaces" by C. Thomassen, which appeared in the American Mathematical Monthly in 1992. It can be found here along with a lot of other fascinating papers, especially if you care about mathematical history. By the way, there's a (fixable) error in Thomassen's paper. See the comment of Robin Chapman to the first answer of the mathoverflow posting here. Finally, another short proof can be found in Doyle-Moran's paper "A short proof that compact $2$-manifolds can be triangulated", whose bibliography info is Invent. Math. 5 1968 160--162.

Lecture 5 : [pdf]
Graphs and trees

Lecture 6 : [pdf]
Classification of surfaces

Homework 3 : [pdf]

Lecture 7 : [pdf]
Klein bottle, manifolds w/ boundary, classification of compact surfaces w/ boundary.

Lecture 8 : [pdf]
Paths and the fundamental group

Homework 4 : [pdf]

Lecture 9 : [pdf]
Functoriality of pi_1, homotopies, Brouwer fixed point theorem

Lecture 10 : [pdf]
Fundamental group of circle

Lecture 11 : [pdf]
Fundamental theorem of algebra, Borsuk-Ulam theorem, contractibility and k-connectivity

Homework 5 : [pdf]

Lecture 12 : coming soon

Lecture 13 : coming soon

Lecture 14 : coming soon

Homework 6 : [pdf]

Lecture 15 : [pdf]
Proof of Seifert van Kampen

Lecture 16 : [pdf]
Fundamental group of graphs

Homework 7 : [pdf]

Lecture 17 : [pdf]
Fundamental groups of CW complexes

Lecture 18 : [pdf]
Intro to knot theory

Lecture 19 : [pdf]
Wirtinger presentation, torus knots

Lecture 20 : [pdf]
Finish torus knots

Homework 8 : none, study for exam next week. It will be an open book, open notes take home exam with a 5 hour time limit. It will be handed out next Friday and will be due the following Wednesday.

Midterm : [pdf]
The midterm is due on Friday, Oct 22 in class. You may print it out at your leisure. The first page contains the instructions. Your time limit begins once you start reading the second page.

Homework 9 : [pdf]

Lecture Notes : I have not posted lecture notes for the stuff on covering spaces. I have been following the treatment in Hatcher's book "Algebraic Topology", which is available on his webpage here. It is Section 1.3 and goes from pages 56-78.

Homework 11 : [pdf]

Homework 12 : [pdf]

"Topological methods in group theory" by Scott and Wall : [pdf]

Homework 13 : [pdf]

Final : [pdf]
The midterm is due on Dec 15th. Please turn it in to me either 1. handing it to me, 2. sliding it under my door, or 3. placing it in my mailbox. You may print it out at your leisure. The first page contains the instructions. Your time limit begins once you start reading the second page.