MT561 Introduction to Algebraic Topology

6 ECTS - 3-0 Duration (T+A)- . Semester- 3 National Credit

Information

Code MT561
Name Introduction to Algebraic Topology
Term 2024-2025 Academic Year
Semester . Semester
Duration (T+A) 3-0 (T-A) (17 Week)
ECTS 6 ECTS
National Credit 3 National Credit
Teaching Language Türkçe
Level Yüksek Lisans Dersi
Type Normal
Mode of study Yüz Yüze Öğretim
Catalog Information Coordinator Prof. Dr. DOĞAN DÖNMEZ


Course Goal / Objective

Classifying topological spaces using algebraic methods.

Course Content

Topological spaces. Fundamental group and its properties. Covering spaces. Higher homotopy groups. Singular homology

Course Precondition

Pre-requisites None

Resources

J. Rotman : Algebraic Topology

Notes

W. Massey Algebraic Topology


Course Learning Outcomes

Order Course Learning Outcomes
LO01 Can define the fundamental group of a topological space
LO02 Can define homotopy and being homotopy equivalent. Knows some consquences of being homotopic.
LO03 Can define higher homotopi groups.
LO04 Knows the homology groups of some simple spaces such as spheres.
LO05 Knows the Eilenberg-McLane Axioms.


Relation with Program Learning Outcome

Order Type Program Learning Outcomes Level
PLO01 Bilgi - Kuramsal, Olgusal Knows in detail the relationship between the results in her area of expertise and other areas of mathematics. 4
PLO02 Bilgi - Kuramsal, Olgusal Knows in detail the relationship between the results in his area of ​​expertise and other areas of mathematics. 4
PLO03 Bilgi - Kuramsal, Olgusal Establishes new mathematical models with the help of the knowledge gained in the field of specialization. 3
PLO04 Bilgi - Kuramsal, Olgusal Has basic knowledge in all areas of mathematics. 4
PLO05 Bilgi - Kuramsal, Olgusal It presents the knowledge gained in different fields of mathematics and their relations with each other in the simplest and most understandable way. 4
PLO06 Bilgi - Kuramsal, Olgusal Effectively uses the technical equipment needed to express mathematics. 3
PLO07 Bilgi - Kuramsal, Olgusal poses original problems related to field and presents different solution techniques. 3
PLO08 Bilgi - Kuramsal, Olgusal carries out original and qualified scientific studies on the subject related to its field.
PLO09 Bilgi - Kuramsal, Olgusal Analyzes existing mathematical theories and develops new theories. 3
PLO10 Beceriler - Bilişsel, Uygulamalı Knows the teaching-learning techniques in areas of mathematics that require expertise and uses these techniques effectively at every stage of education. 3
PLO11 Yetkinlikler - Bağımsız Çalışabilme ve Sorumluluk Alabilme Yetkinliği To have knowledge of a foreign language at a level to be able to follow foreign sources related to the field and to communicate verbally and in writing with foreign stakeholders. 3
PLO12 Yetkinlikler - Bağımsız Çalışabilme ve Sorumluluk Alabilme Yetkinliği presents and publishes its original works within the framework of scientific ethical rules for the benefit of its stakeholders. 3
PLO13 Yetkinlikler - Öğrenme Yetkinliği Adheres to the ethical rules required by its scientific title 5


Week Plan

Week Topic Preparation Methods
1 Homotopy between maps and homtopic spaces. Contractible spaces. Studying the relevant parts of the course materials. Öğretim Yöntemleri:
Anlatım
2 Definition and basic properties of the fundamental group. Studying the relevant parts of the course materials. Öğretim Yöntemleri:
Anlatım
3 Functorial properties of the fundamental group. Relation with homotopy. Studying the relevant parts of the course materials. Öğretim Yöntemleri:
Anlatım
4 Covering spaces and the Covering Homotopy Property. Studying the relevant parts of the course materials. Öğretim Yöntemleri:
Anlatım
5 Relationship between covering spaces and the fundamental group. Studying the relevant parts of the course materials. Öğretim Yöntemleri:
Anlatım
6 Relationship between covering spaces and homotopy groups Studying the relevant parts of the course materials. Öğretim Yöntemleri:
Anlatım
7 Affine space, standart and singular simplexes. Chain groups Studying the relevant parts of the course materials. Öğretim Yöntemleri:
Anlatım
8 Mid-Term Exam Solve the homework problems. Ölçme Yöntemleri:
Ödev
9 Face operators and bounday map. Studying the relevant parts of the course materials. Öğretim Yöntemleri:
Anlatım
10 Singular chain complex and its properties. Studying the relevant parts of the course materials. Öğretim Yöntemleri:
Anlatım
11 Relative Homology groups. Long exact sequence. Studying the relevant parts of the course materials. Öğretim Yöntemleri:
Anlatım
12 Homology groups. Homology functors. Studying the relevant parts of the course materials. Öğretim Yöntemleri:
Anlatım
13 Homotopy invariance property. Studying the relevant parts of the course materials. Öğretim Yöntemleri:
Anlatım
14 Excision property. Studying the relevant parts of the course materials. Öğretim Yöntemleri:
Anlatım
15 Eilenberg-McLane Axioms Studying the relevant parts of the course materials. Öğretim Yöntemleri:
Anlatım
16 Term Exams Solve the homework problems. Ölçme Yöntemleri:
Ödev
17 Term Exams Solve the homework problems. Ölçme Yöntemleri:
Ödev


Student Workload - ECTS

Works Number Time (Hour) Workload (Hour)
Course Related Works
Class Time (Exam weeks are excluded) 14 3 42
Out of Class Study (Preliminary Work, Practice) 14 5 70
Assesment Related Works
Homeworks, Projects, Others 0 0 0
Mid-term Exams (Written, Oral, etc.) 1 15 15
Final Exam 1 30 30
Total Workload (Hour) 157
Total Workload / 25 (h) 6,28
ECTS 6 ECTS

Update Time: 09.05.2024 11:37