# Algebraic Topology 1

Welcome to the course page of *Algebraic Topology 1* at SNU (Fall 2018). You can enter this page at http://cayley.kr/wiki/at1

## Contents

## Basic Info

- Course
- SNU 3341.607 (001) Algebraic Topology 1, Fall 2018

- Web
- 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 3:30 - 4:45pm, Bldg 24-211

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

- TA
- 김재형
- mlrde01(aht)snu

### 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%)

Late submission is not accepted.

- Midterm (30%)
- October 24th (W), during the class

- Final Exam (40%)
- December 5th (W), 3:30 - 5 pm.

### Textbook

- First course in Topology, Part II:
**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 Homotopy, fundamental groups, base change

Week 2 Homotopy invariance, covering space

Week 3 Lifting theorems, deck transformation, covering--subgroup correspondence

Week 4

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

Week 5 Homology groups, homotopy invariance

Week 6 Relative homology groups, long exact sequence

Week 7 Excisions

Week 8 Homology of spheres and their generators, Brouwer Fixed Point Theorem, Fundamental Theorem of Algebra, Local homology and Invariance of Dimension

Week 9
Hurewicz Theorem for π_{1}, Homology of surfaces

Week 10 CW complex, degree, cellular homology

Week 11 Mayer--Vietoris sequence, Computations

Week 12 Jordan–Brouwer Separation Theorem

Week 13 Invariance of Domain, Borsuk--Ulam Theorem, Simplicial Approximation Theorem, Lefschetz Fixed Point Theorem

## 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.