**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:**

- Adam Block abb2190@columbia.edu
- Quang Dao qvd2000@columbia.edu

**TA Office Hours:**

**References:**

- A. Hatcher,
. Free download is available here.*Algebraic Topology* - M. Armstrong,
*Basic Topology*

*Some other relevant books:*

- V. A. Vassiliev "Introduction to Topology"
- James Munkres "Elements of Algebraic Topology"
- James Munkres "Topology"

**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 |