Math W4053: Intro to Algebraic Topology
Email: hom [at] math.columbia.eduLocation: 520 Math
Time: TR 1:10-2:25
Office: 613 Math
Office hours: Tuesdays 2:30-3:30 pm, Wednesdays 11 am-noon
TA: Sara Venkatesh
TA help room hours: Wednesdays 4-6 pm, Thursdays 9-10 am in 406 Math
Textbook
Algebraic Topology, A. Hatcher. Free download; a print version is available for about $40. The plan is to cover Chapters 0-2: fundamental group, covering spaces, homology, and other topics as time permits.Homework
Homework will count for 30% of your grade and is due on Thursdays at the beginning of class in the box outside of 417 Math. Your lowest homework score will be dropped.Exams
There will be three midterms. The first two will be in class on Tuesday, February 17 and Tuesday, March 31, each worth 20% of your grade. The third midterm will be a take-home exam, due on Monday, May 4, worth 30% of your grade. There will be no make-up exams.Help
If you find yourself confused, please seek help sooner rather than later. I will be available to answer questions during my office hours. Additionally, there is the Columbia help room in 406 Math.In order to receive disability-related academic accommodations, students must first be registered with the Disability Services (DS). More information on the DS registration process is available online at www.health.columbia.edu/ods. Registered students must present an accommodation letter to the professor before exam or other accommodations can be provided. Students who have, or think they may have, a disability are invited to contact DS for a confidential discussion.
Schedule & Assignments
Prerequisites: Math V21010 (Linear Algebra), Math W4041 (Modern Algebra I), Math W4051 (Topology)The schedule below is tentative and subject to change.
| Date | Material (from Hatcher) | Homework |
| 1/20 | Introduction, homotopy, operations on spaces (Chapter 0) | |
| 1/22 | CW complexes (Chapter 0) | HW 1 due Thurs Jan 29 |
| 1/27 | Snow Day | |
| 1/29 | More on CW complexes (Chapter 0) | HW 2 due Thurs Feb 5 |
| 2/3 | Paths, homotopy, and the fundamental group (Section 1.1) | |
| 2/5 | The fundamental group of the circle (Section 1.1) | HW 3 due Thurs Feb 12 |
| 2/10 | Induced homomorphisms, applications (Section 1.1) | |
| 2/12 | Seifert-van Kampen's theorem (Section 1.2) (not on Midterm I) | Study for Midterm I |
| 2/17 | Midterm I (in class) | |
| 2/19 | Applications of Seifert-van Kampen's theorem (Section 1.2) | HW 4 due Thurs Feb 26 |
| 2/24 | Covering spaces (Section 1.3) | |
| 2/26 | Covering spaces (Section 1.3) | HW 5 due Thurs Mar 5 |
| 3/3 | Simply connected covering spaces (Section 1.3) | |
| 3/5 | Classification of covering spaces (Section 1.3) | HW 6 due Thurs Mar 12 |
| 3/10 | Actions on spaces (Section 1.3) | |
| 3/12 | Simplicial homology (Section 2.1) | HW 7 due Thurs Mar 26 |
| 3/17 | Spring Break | |
| 3/19 | Spring Break | |
| 3/24 | Singular homology (Section 2.1) | |
| 3/26 | Homotopy invariance, exact sequences (Section 2.1) | Study for Midterm II |
| 3/31 | Midterm II (in class) | |
| 4/2 | Long exact sequences (Section 2.1) | HW 8 due Thurs Apr 9 |
| 4/7 | Relative homology, excision (Section 2.1) | |
| 4/9 | Equivalence of simplicial and singular homology (Section 2.1) | HW 9 due Thurs Apr 16 |
| 4/14 | Degree, cellular homology (2.2) | |
| 4/16 | Euler characteristic, Mayer-Vietoris sequence (2.2) | HW 10 due Thurs Apr 23 |
| 4/21 | Homology with coefficients (2.2), axioms for homology (2.3) | |
| 4/23 | Homotopy groups (4.1) Midterm III handed out at end of class |
|
| 4/28 | Category theory (2.3), student presentations (Jincheng) | |
| 4/30 | Student presentations (Lena, Robbie) Note: NO final exam for this class |
Midterm III (take home) due Mon 5/4 |