Math GU4053: Introduction to Algebraic Topology

Email: alishahi@math.columbia.edu
Time and Place: Tuesday-Thursday: 2:40pm-3:55pm at Math 312
Course webpage: http://www.math.columbia.edu/~alishahi/IntroAlgTop.html
Office hours: Tuesday 4:30pm-6:30pm ; in Math 613
Teaching Assistant:

TA Office Hours:


References:

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 Thursdays. If you can't make it to the class, put it in the assigned box outside of 417 Math. The lowest homework score will be dropped.

Homework 1
Homework 2
Homework 3
Homework 4
Homework 5
Homework 6
Homework 7
Homework 8
Homework 9
Homework 10
Homework 11


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 15
Lecture 16
Lecture 17
Lecture 18
Lecture 19
Lecture 20
Lecture 21
Lecture 22
Lecture 23
Lecture 24
Lecture 25
Lecture 26


Exam: There will be two midterm exam in class, and a take-home final exam.

Midterm 1: February 19
Midterm 2: March 26
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 Topics References
1/22 Introduction, CW complexes. Overview of Chapter 5 Sections 1-4 of Armstrong and pages 5-7 of Hatcher
1/24 Continue with CW complex examples, Fee product of groups, Van Kampen's theorem: statement, examples Pages 7-8 and 40-44 of Hatecher, Appendix of Armstrong
1/29 Application: Computing fundamental group of CW complexes Section 1.2
1/31 Surfaces Sections 7.1 and 7.5 of Armstrong
2/5 Classification of surfaces Continue with 7.5 and 7.3 of Armstrong
2/7 Covering spaces: definitions, examples. Lifting lemmas. Section 1.3
2/12 Covering spaces(Continued) Section 1.3
2/14 Classification of covering spaces Section 1.3
2/19 Midterm 1
2/21 Actions on covering spaces Section 1.3
2/26 Triangulation and Delta-complexes Section 2.1
2/28 Simplicial homology Section 2.1
3/5 Singular homology Section 2.1
3/7 Homotopy invariance, exact sequences Section 2.1
3/12 Long exact sequences Section 2.1
3/14 Relative homology, excision Section 2.1
3/19 Spring break
3/21 Spring break
3/26 Midterm 2
3/28 Equivalence of simplicial and singular homology Section 2.1
4/2 Euler characteristic, Mayer-Vietoris sequence Section 2.2
4/4 Cellular homology. Introduction to degree theory Section 2.2
4/9 Homology with coefficients, axioms for homology Sections 2.2 and 2.3
4/11 Cohomology: definition, examples Section 3.1
4/16 Cohomology: basic properties Section 3.1
4/18 Universal coefficient theorem Section 3.1
4/23 Cup product: definition, examples. Section 3.2
4/25 More on Cup product, cohomology ring Section 3.2
4/30 Orientability and the fundamental class Section 3.3
5/2 Poincare Duality Section 3.3