Algebraic Topology 1, Fall 2016

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

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.
Facebook
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

References

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 π1

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 π1, 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

Due on September 21
Due on October 19


Due on November 2
Due on November 16
Due on November 30

Pictures

Midterm Exam

Definitions

Problems

Solutions

Final Exam

Problems

Solutions