Math 6441

Algebraic Topology

Spring 2019
-
----------------------------------------M.C. Escher

Class Meetings

 Monday, Wednesday, Friday, 9:05-9:55 am, Skiles 311.

Book

Notes

 The lectures will be based on these notes.

Office Hours

 In Skiles 234, after class and by appointment.

Homework

 There will be daily homework assignments, each with 1-3 problems. Homework is due at the start of the next class. The three lowest scores will be dropped. You should resist the urge to Google the answers.

Grading

 Course grades will be determined by homework scores.

Weekly Schedule
Week Dates Topics Text Sections Homework Lectures Notes
1 Jan 7 Intro / Spaces Chapter 0 p. 18 #1,2,3 16,20
2 Jan 14 Fundamental group Section 1.1 p. 18 #23 p. 38 #6 #7,8,9
3 Jan 21 Van Kampen's theorem Section 1.2 p. 38 #16 p. 38 #19 MLK day
4 Jan 28 Covering spaces Section 1.3 p. 52 #1,3,4 #8,9,11 p. 79 #1,4,10
5 Feb 4 Classifying spaces Section 1.3 p. 79 #9 #12,14 18,19
6 Feb 11 Homology Section 2.1 p. 96 #1,2 p.131 #4,5 #7,8,9
7 Feb 18 Relative homology Section 2.1 p. 131 # 11,12,13 #17,21 #22
8 Feb 25 Excision / Mayer-Vietoris Section 2.1 p. 131 # 28 # 29 p. 176 #2,6
9 Mar 4 Applications Section 2.1 p. 176 #8 p. 184 #5 p. 155 #8,22,23,24
10 Mar 11 Cohomology Section 2.1 - p. 204 # 5,6 #10
Spring Break
11 Mar 25 Cup product Section 2.2 - p. 204 #8,9 p. 228 #1,3,7
12 Apr 1 Poincaré duality Section 3.1 - p. 228 #10,11,12 p. 257 #6,24,25 Cup & Cap
13 Apr 8 Spectral sequences Section 3.2 p. 257 #10,11,26 below Spectral Sequences
14 Apr 15 Spectral sequences Section 3.2 below below
15 Apr 22 Spectral sequences Section 3.3

Additional homework

Resources

 ---Algebraic Topology, Allen Hatcher

---Lecture Notes in Algebraic Topology, James F. Davis and Paul Kirk

---Algebraic Topology Class Notes, Denis Sjerve (notes by Benjamin Young)

---Algebraic Topology I, Tim Perutz

---algebraic topology, Michael Hopkins (notes by Eva Belmont and Akhil Mathew)

---Algebraic Topology, Len Evans and Rob Thompson

---Spectral Sequences in Algebraic Topology, Allen Hatcher

---Introduction to Spectral Sequences, Michael Hutchings

---Lyndon-Hochschild-Serre Spectral Sequence - Application, Benjamin Ruppik

---The topology of fiber bundles, Ralph L. Cohen

---Spectral sequences via examples, Antonio Díaz Ramos

---Homology: An idea whose time has come, Barry A. Cipra

---Assembling geometric data using statistics for topology, Peter Bubenik

---Barcodes: The Persistent Topology of Data, Robert Ghrist

---Persistent Homology - a Survey, Herbert Edelsbrunner and John Harer

---Computing Persistent Homology, Afra Zomorodian and Gunnar Carlsson

---A History of Duality in Algebraic Topology, James Becker and Daniel Gottlieb

---Visualizing Poincare Duality, Lucien Clavier

---On proof and progress in mathematics, William Thurston

---Proof that R3 cannot be made into a field

---Georgia Tech Honor Code