Math 4432

(Algebraic) Topology

Spring 2015

------------------------------J.J. van Wijk & A. Cohen

News: Welcome to Math 4432!

Professor

Textbook

Class Meetings

Monday, Wednesday, and Friday, 11:05-11:55 pm, Skiles 246.

Office Hours

Wednesdays 3-4, Thursdays 10-11, after class, and by appointment.

Course Objectives

The goal of this course is to gain basic familiarity with the objects of low-dimensional topology - knots, surfaces, and 3-manifolds - as well as some of the basic tools used to study them, for instance, Euler characteristic and fundamental groups. We will emphasize proofs and problem solving alongside computation.

Homework

Homework will be assigned on a weekly basis. Homework assigned during a given week is due by 11:05 am on the Friday of the following week. Homework will be submitted online via Piazza.

Grading

Homework 10%, Midterms 20%, Final Exam 30%.

Daily Schedule

Week	Date	Topics	Reading	Homework	Notes
1	Jan 5	Overview		Autobiography
	Jan 7	Topological equivalence	1.4
	Jan 9	Topological equivalence	1.4	1.4 # 8, 9, 10, 12, 15	Last day to drop w/o W
2	Jan 12	Topological invariants	1.5
	Jan 14	Topological invariants	1.5	1.5 # 7, 10, 12, 13
	Jan 16	Isotopy	1.6	1.6 # 4, 5, 7, 9
3	Jan 19				MLK Day
	Jan 21	Knots	2.1	2.1 # 7, 9b, 10
	Jan 23	Knot diagrams	2.2	2.2 # 11, 12, 14, 15
4	Jan 26	Reidemeister moves	2.3	2.3 # 3, 10, 11
	Jan 28	Colorings	2.4	2.4 # 1, 5
	Jan 30	First Midterm	1.4-1.6, 2.1-2.4		Solutions
5	Feb 2	Alexander polynomial	2.5	2.4 # 8
	Feb 4	Alexander polynomial	2.5	2.5 # 1,3,14,16
	Feb 6	Skein relations	2.6	2.6 # 6,7,9,11
6	Feb 9	Jones polynomial	2.7	2.7 # 3,10,11
	Feb 11	Surfaces	3.1	3.1 # 2, 3, 7, 10
	Feb 13	Cutting and pasting	3.2	3.2 # 1, 4, 6, 8
7	Feb 16	Euler characteristic	3.3	3.3 # 1, 4
	Feb 18	Classification of surfaces	3.4	3.4 # 3, 4, 11
	Feb 20	Seifert surfaces	3.5	3.5 # 1, 2
8	Feb 23	Mapping class groups	OHGGT	Read / make comments
	Feb 25				Snow day
	Feb 27	Mapping class groups	OHGGT	Read / make comments	Last day to drop with W Last day to elect P/F
9	Mar 2	3-manifolds	4.1	4.1 # 5, 7, 8
	Mar 4	Euler characteristic	4.2	4.2 # 3, 4, 5, 6, 9
	Mar 6	Second Midterm	2.5-2.7, 3.1-3.5
	Mar 9	Gluing polyhedra	4.3	4.3 # 2, 5, 6, 9
	Mar 11	Heegaard splittings	4.4	4.4 # 7
	Mar 13	Heegaard splittings	4.4	4.4 # 11
	Mar 16				Spring break
11	Mar 23	Fixed points in intervals Sharkovsky's theorem	5.1	5.1 # 8, 9, 10
	Mar 25	Contraction mapping theorem	5.2	5.2 # 2, 3, 7
	Mar 27	Sperner's lemma	5.3	5.3 # 2
12	Mar 30	Brouwer FPT	5.4	5.4 # Prove Ham Sandwich Thm
	Apr 1	Fundamental group	6.1	6.1 # 5
	Apr 3	Fundamental group	6.1	6.1 # 6
13	Apr 6	Algebraic properties	6.2	6.2 # 2
	Apr 8	Invariance	6.3	6.3 # 3, 6, 8
	Apr 10	Third Midterm	4.1-4.4, 5.1-5.4
14	Apr 13	Spheres	6.4	6.4 # 9, 10, 11, 12, 13, 15, 18
	Apr 15	Words and relations	6.5	6.5 # 5, 7, 10
	Apr 17	Words and relations	6.5	6.5 # 14
15	Apr 20	Covering spaces	6.5
	Apr 22	Covering spaces	6.5
	Apr 24	Quintics	Notes
16	Apr 29	Final Exam	1.4-1.6, 2.1-2.7, 3.1-3.5, 4.1-4.4, 5.1-5.4, 6.1-6.6

References

---Mathematical Omnibus: Thirty Lectures on Classic Mathematics, Dmitry Fuchs and Serge Tabachnikov

---The topological proof of Abel-Ruffini theorem, Henryk Zoladek

---The Sharkovsky Theorem: A Natural Direct Proof, Keith Burns and Boris Hasselblatt

---The Knot Book, Colin Adams

---Teruaki, Kazushi Ahara

---Topology, James Munkres

---The Shape of Space, Jeffrey Weeks

---Understanding Topology and Shapefiles, David M. Theobald www.math.psu.edu/tabachni/Books/taba.pdf

Resources

---T-Square

---Georgia Tech Honor Code