# 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

## 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.
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.

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.

## 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 π1

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 π1, 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 No. 1

Please refer to the eTL bulletin board.

### Exercises

• You can find good exercise problems in SNU Graduate Entrance and SNU PhD Qualifying
• Students are strongly recommended to work together on problems. Writing must be done by oneself.
• Late HW policy: not accepted.