Email: alishahi@math.columbia.edu
Time and Place: Monday-Wednesday: 4:10pm-5:25pm at Math 417
Course webpage: http://www.math.columbia.edu/~alishahi/IntroAlgTop.html
Office hours: Tuesday:noon-1pm and Wednesday: 5:30pm-6:30pm or by appointment; in Math 613
Teaching Assistant: James Cornish, cornish@math.columbia.edu
TA Office Hours: TWR: noon-1pm (Math 406)
Textbook: Allen Hatcher, Algebraic Topology. Free download is available here.
Some other relevant books:
Prerequisite: A background in point-set topology (e.g., Math GU4051) and abstract algebra (e.g., Math GU4041).
Homework: There will be problem sets every week, due at the beginning of class on Mondays. If you can't make it to the class, put it in the assigned box outside of 417 Math. The lowest homework scores will be dropped.
Homework 1
Homework 2
Homework 3
Homework 4
Homework 5
Homework 6
Homework 7
Homework 8
Homework 9
Homework 10
Class Notes:
Lecture 1
Lecture 2
Lecture 3
Lecture 4
Lecture 5
Lecture 6
Lecture 7
Lecture 8
Lecture 9
Lecture 10
Lecture 11
Lecture 12
Lecture 13
Lecture 14
Lecture 16
Lecture 17
Lecture 18
Lecture 19
Lecture 20
Lecture 21
Lecture 22
Exam: There will be two midterm exam in class, and a take-home final exam.
Midterm 1: February 13
Midterm 2: March 22
Final: Take-home
There will be no make-up exams.
Grading: The final course grade will be determined by:
Homework: 30%
Midterm 1: 20%
Midterm 2: 20%
Final: 30%
Getting help. If you're having trouble, get help immediately. I will be available to answer questions during my office hours. Additionally, there is the Columbia help room in 406 Math.
Student with disabilities: Students must register with the Disability Services and present an accomodation letter before the exam or other accomodations that can be provided. More information is available on the Disability Services webpage.
Tentetive Schedule:
| Date | Sections | Textbook |
|---|---|---|
| 1/18 | Introduction, Homotopy and homotopy equivalence. Operations on spaces. | Chapter 0 |
| 1/23 | CW complexes. Some familiar spaces. | Chapter 0 |
| 1/25 | Paths, homotopy, and the fundamental group | Section 1.1 |
| 1/30 | The fundamental group of the circle. Applications. | Section 1.1 |
| 2/1 | Van Kampen's theorem: statement, examples | Section 1.2 |
| 2/6 | Applications of Seifert-van Kampen's theorem | Section 1.2 |
| 2/8 | Covering spaces: definitions, examples. Lifting lemmas. | Section 1.3 |
| 2/13 | Midterm 1 | |
| 2/15 | Covering spaces(Continued) | Section 1.3 |
| 2/20 | Classification of covering spaces | Section 1.3 |
| 2/22 | Actions on covering spaces | Section 1.3 |
| 2/27 | Simplicial homology | Section 2.1 |
| 3/1 | Singular homology | Section 2.1 |
| 3/6 | Homotopy invariance, exact sequences | Section 2.1 |
| 3/8 | Long exact sequences | Section 2.1 |
| 3/13 | Spring break | |
| 3/15 | Spring break | |
| 3/20 | Relative homology, excision | Section 2.1 |
| 3/22 | Midterm 2 | |
| 3/27 | Euler characteristic, Mayer-Vietoris sequence | Section 2.2 |
| 3/29 | Applications | |
| 4/3 | Cellular homology. Introduction to degree theory | Section 2.2 |
| 4/5 | Cohomology: definition, examples | Section 3.1 |
| 4/10 | Cohomology: basic properties | Section 3.1 |
| 4/12 | Universal coefficient theorem | Section 3.1 |
| 4/17 | Künneth formula | Section 3.2 |
| 4/19 | Cup product: definition, examples. | Section 3.2 |
| 4/24 | Orientability and the fundamental class | Section 3.3 |
| 4/26 | Poincaré duality | Section 3.3 |
| 5/1 | More on Duality | Section 3.3 |