Basic Topology

Course Code: B12003Y

Course Name: Basic Topology

Credits: 4.0

Level: Undergraduate

Pre-requisite: Null

Lecture Time: 20 weeks, 2 sessions/week, 2 hours/session

Instructors:Null

Course Description

This course is designed for undergraduate students with major in mathematical sciences and is also one of the required courses for other physics majors. The main contents of this course include: (1)Point set topology；(2) Combinatorial topology (the origin of algebraic topology); (3) Algebraic topology.

Since topology is a branch of geometry, the aim of this class is to improve students’ ability of geometric imagination on geometry and their ability to analyze geometry problems. In particular, the teacher will introduce the main characteristics and methods of topological space together with the basic theory of topology to lay foundation for future courses.

Topics and Schedule

  1. Introduction

1.1.Euler theorem

1.2.Topological equivalence

1.3.Surface

1.4.Abstract space

1.5.Feit-Thompson theorem

1.6.Invariants

  1. Topological space and continuous map

2.1.Open sets and closed sets

2.2.Continuous map

2.3.Space-filling curve

2.4.Tietze extension theorem

  1. The convergence and connectedness of topological space

3.1.Closed sets in Euclidean space

3.2.Heine Borel theorem

3.3.Convergence space

3.4.Multiple spaces

3.5.Path connectedness

  1. Gluing space

4.1.The creation of the Mobius strip

4.2.Gluing topology

4.3.Topological group

4.4.Orbit space

  1. Topological groups

5.1.Homotopy map

5.2.Topological groups

5.3.Calculus

5.4.Homotopy equivalence

5.5.Brouwer fixed-point theorem

5.6.Plane separation

5.7.The boundary of surface

5.8.Covering space

  1. Simplicialsubdivision

6.1.Simplicial subdivision of spaces

6.2.Barycentric subdivision

6.3.Simplicial approximation

6.4.The Homotopy groups of complexes

6.5.Simplicial division of Orbit space

  1. Surface

7.1.Classification

7.2.Simplex division and Orientation

7.3.Euler characteristic

7.4.Calculation of digging and adding

7.5.Surface symbol

  1. Simplicial homology

8.1.Closed line and boundary

8.2.Homology group

8.3.Examples

8.4.Simplicial map

8.5.Redivision

8.6.Invariance

  1. Mapdegreeand Lefschetz number

9.1.The continuous map of sphere

9.2.Euler Poincaré equation

6.3.Borsuk Ulam theorem

6.4.Lefschetz fixed-point theorem

Grading

The grades include midterm examination, final examination, and assignments of weekly homework.

Textbook

Armstrong, Mark Anthony. Basic topology. Springer Science & Business Media(Chinese Version), translated by Yifeng Zhang.

References

[1]Chengye You, Basic topology , Beijing University Press, 1997 (Chinese Version)

Course Website

To be added