# An Introduction to Low Dimensional Topology (MATH577) Course Detail

Course Name Course Code Season Lecture Hours Application Hours Lab Hours Credit ECTS
An Introduction to Low Dimensional Topology MATH577 3 0 0 3 5
Pre-requisite Course(s)
Consent of the Department
Course Language English N/A Natural & Applied Sciences Master's Degree Face To Face Discussion, Question and Answer. The aim of this course is to introduce the basic techniques to work with manifolds in low dimensions. To approach this main goal, the knot theory and surfaces, which touches upon many branches of mathematics, will be introduced. Elementary linear algebra and some group theory which are taught in the undergraduate program are sufficient as a background. The students who succeeded in this course; learn knots, links and their invariants understand how knots and links are related to the two and three dimensional manifolds learn the braids, their relations to knots and links and Seifert surfaces also to mapping class groups learn the fundamental tools of three manifolds such as Heegaard decompositions, surgery, branched coverings etc. Knots, links and their invariants, Seifert surfaces, braids, mapping class groups, Heegaard decompositions, lens spaces and surface homeomorphisms, surgery of 3-manifolds, branched coverings.

### Weekly Subjects and Releated Preparation Studies

Week Subjects Preparation
1 Knots, Links and Ribbons pp. 1-22
2 Knot and Link Invariants pp. 23-46
3 Seifert Matrices and Surfaces pp. 118-144, pp.200-232 (D.Rolfsen)
4 Braids pp. 47-66
5 3-Manifolds pp.67-72
6 Heegaard Decompositions of 3- Manifolds pp. 73-76
7 Midterm
8 Lens Spaces pp. 77-82
9 Homeomorphisms of Surfaces pp.83-89
10 Mapping Class Groups pp. 90-94
11 Fibred Knots and Links Open Book Decompositions pp.323-341 (D.Rolfsen)
12 Surgery on 3-Manifolds pp. 95-107
13 Surgery on 3-Manifolds (Continue) pp. 108-126
14 Branched Coverings pp. 127-151
15 Review
16 Final Exam

### Sources

Course Book 1. and 3-Manifolds: An Introductions to the New Invariants in Low-Dimensional Topology, Prasolov, Sossinsky, 1996, AMS.

### Evaluation System

Attendance/Participation - -
Laboratory - -
Application - -
Field Work - -
Special Course Internship - -
Quizzes/Studio Critics - -
Homework Assignments 3 30
Presentation - -
Project - -
Report - -
Seminar - -
Midterms Exams/Midterms Jury 1 30
Final Exam/Final Jury 1 40
Toplam 5 100
 Percentage of Semester Work 60 40 100

### Course Category

Core Courses X

### The Relation Between Course Learning Competencies and Program Qualifications

# Program Qualifications / Competencies Level of Contribution
1 2 3 4 5
1 Has the ability to apply scientific knowledge gained in the undergraduate education and to expand and extend knowledge in the same or in a different area.
2 Has the ability to obtain, to evaluate, to interpret and to apply information by doing scientific research.
3 Can apply gained knowledge and problem solving abilities in inter-disciplinary research.
4 Has the ability to work independently within research area, to state the problem, to develop solution techniques, to solve the problem, to evaluate the obtained results and to apply them when necessary.
5 Takes responsibility individually and as a team member to improve systematic approaches to produce solutions in unexpected complicated situations related to the area of study.
6 Can develop strategies, implement plans and principles on the area of study and can evaluate obtained results within the framework.
7 Can develop and extend the knowledge in the area and to use them with scientific, social and ethical responsibility.
8 Has the ability to follow recent developments within the area of research, to support research with scientific arguments and data, to communicate the information on the area of expertise in a systematically by means of written report and oral/visual presentation.
9 Has proficiency in English language and has the ability to communicate with colleagues and to follow the innovations in mathematics and related fields.
10 Has software and hardware knowledge in the area of expertise, and has proficient information and communication technology knowledge.
11 Has professional ethical consciousness and responsibility which takes into account the universal and social dimensions in the process of data collection, interpretation, implementation and declaration of results in mathematics and its applications.

Activities Number Duration (Hours) Total Workload
Course Hours (Including Exam Week: 16 x Total Hours)
Laboratory
Application
Special Course Internship
Field Work
Study Hours Out of Class
Presentation/Seminar Prepration
Project
Report
Homework Assignments 3 5 15
Quizzes/Studio Critics
Prepration of Midterm Exams/Midterm Jury 1 10 10
Prepration of Final Exams/Final Jury 1 10 10