# Algebraic Topology 1, Fall 2016

Welcome to the course page of *Algebraic Topology 1* at SNU (Fall 2016). You can enter this page at http://cayley.kr/wiki/at1

## Contents

## Basic Info

- Course
- SNU 3341.607 (001) Algebraic Topology 1, Fall 2016

- Web
- http://cayley.kr/wiki/at1
- Students are expected to visit this course webpage at least once a week.

- https://www.facebook.com/groups/1078385582238354
- Online discussion is strongly encouraged.

- Classes
- MW 9:30 - 10:45, Bldg 500-306

- Office hour
- MW 3:15 - 4:30, 27-414.

- Instructor
- 김상현 Sang-hyun Kim
- s.kim(aht)snu.ac.kr

- TA
- 이돈성
- disturin(aht)snu.ac.kr

### Topics and Prerequisites

This course is divided into two parts. In the first (short) part, we will review fundamental groups and the functorial properties of them. The topics include covering spaces, Seifert–van Kampen theorem, graphs of groups and K(G,1) spaces.

The second (main) part will mainly deal with homology theory. We will talk about functorial property, computation through simplicial homology and Mayer–Vietoris sequence and relation to fundamental groups.

Prerequisites are point-set topology, linear algebra, algebra and topology at undergraduate levels.

### Requirements and grading

- Attendance and Participation (10%)
- Problem Sets (20%)

Due on Wednesday 5 pm. Late submission is not accepted.

- Midterm (30%)
- October 19th (W), during the class
- Final Exam (40%)
- December 5th, 9 - 11 am (M).

### Textbook

- First course in Topology:
**being updated**every week! - Hatcher, Alan, Algebraic Topology, http://www.math.cornell.edu/~hatcher/AT/ATpage.html
- Bredon, G. E. (1997). Topology and geometry (Vol. 139, pp. xiv–557). Springer-Verlag, New York.

### References

- Massey, William S., Algebraic Topology: An Introduction, Graduate Texts in Mathematics, Springer-Verlag, New York, 1977.
- 김혁 교수님 강의록, http://mathlab.snu.ac.kr/~tl
- Conway's ZIP proof

## Lecture Plan

Subject to change.

Week 1 (September 5, 7) Homotopy, fundamental groups, base change

Week 2 (September 12) Homotopy invariance, covering space

Week 3 (September 19, 21)

- 19 : Lifting theorems
- 21 : applications, deck transformation, covering--subgroup correspondence

Week 4 (September 26)

- 26 : Seifert--van Kampen theorem, surface π
_{1}

Week 5 (October 5) Homology groups, homotopy invariance

Week 6 (October 10, 12) Relative homology groups, long exact sequence

Week 7 (October 17, 19) Excisions

Week 8 (October 24, 26) Homology of spheres and their generators, Brouwer Fixed Point Theorem, Fundamental Theorem of Algebra, Local homology and Invariance of Dimension

Week 9 (October 31, November 2)
Hurewicz Theorem for π_{1}, Homology of surfaces, Midterm Exam

Week 10 (November 7, 9) CW complex, degree, cellular homology

Week 11 (November 14, 16) Mayer--Vietoris sequence, Computations

Week 12 (November 23) Jordan–Brouwer Separation Theorem

Week 13 (November 28, 30) Invariance of Domain, Borsuk--Ulam Theorem, Simplicial Approximation Theorem, Lefschetz Fixed Point Theorem

- December 5
- Final Exam

## Homework

- HW source: https://www.dropbox.com/s/y6xq6bxqbxa0fbf/shkim-top1note-main.pdf?dl=0
- You can find good exercise problems in SNU Graduate Entrance and SNU PhD Qualifying
- Use the HW box in TBA.
- Due on Wednesday, 5 pm.
- Please pick up your graded HW one week later, at the TA office.
- Students are
*strongly*recommended to work together on problems. Writing must be done by oneself. - Late HW policy: not accepted.

- Due on September 21

- Due on October 19

- Due on November 2

- Due on November 16

- Due on November 30