ALGEBRAIC TOPOLOGY
| Title | ΑΛΓΕΒΡΙΚΗ ΤΟΠΟΛΟΓΙΑ / ALGEBRAIC TOPOLOGY |
| Code | Θ010 |
| Faculty | Sciences |
| School | Mathematics |
| Cycle / Level | 2nd / Postgraduate |
| Teaching Period | Winter/Spring |
| Coordinator | Georgios Raptis |
| Common | Yes |
| Status | Active |
| Course ID | 600025890 |
Programme of Study: PMS Tmīmatos Mathīmatikṓn (2025-2030)
Registered students: 0
| Orientation | Attendance Type | Semester | Year | ECTS |
|---|---|---|---|---|
| THEŌRĪTIKA MATHĪMATIKA | Elective Courses belonging to the selected specialization | Spring | - | 10 |
| Academic Year | 2025 – 2026 |
| Class Period | Spring |
| Faculty Instructors | |
| Weekly Hours | 3 |
| Total Hours | 39 |
| Class ID | 600268416
|
Type Of Offer
- Disciplinary Course
Course Type 2021
Specialization / Direction
Mode of Delivery
- Face to face
- Distance learning
Digital Course Content
- e-Study Guide https://qa.auth.gr/en/class/1/600268416
Erasmus
The course is also offered to exchange programme students.
Language of Instruction
- Greek (Instruction, Examination)
- English (Instruction, Examination)
Prerequisites
General Prerequisites
Basic knowledge of general topology and group theory at the undergraduate level
Learning Outcomes
Upon successful completion of the course, students will:
1. know concepts and applications of algebraic topology, which are of great significance for modern pure mathematics
2. be familiar with applications of algebraic topology in other fields
3. acquire knowledge of category theory and its uses in topology and geometry
4. be familiar with the homotopy-theoretic approach to the study of spaces
5. know how to compute the fundamental group and other homotopical invariants
6. have learned central homotopical properties of important spaces such as spheres, projective spaces and surfaces
General Competences
- Make decisions
- Work autonomously
- Work in teams
- Generate new research ideas
- Be critical and self-critical
- Advance free, creative and causative thinking
Course Content (Syllabus)
Topological spaces. Constructions with topological spaces. Spheres, surfaces, and projective spaces. Categories, functors and natural transformations. Homotopy, homotopy equivalences. The fundamental group. Covering spaces. Classification of covering spaces. Calculations of fundamental groups. The Seifert-van Kampen theorem. Applications: the Brouwer fixed point theorem and the fundamental theorem of algebra. Basic notions of the homotopy theory of topological spaces (cofibrations, fibrations, Puppe sequences, homotopy groups, suspensions, loops, etc.). CW-complexes. Whitehead's theorem. CW-approximation. Eilenberg-Mac Lane spaces. The Blakers-Massey homotopy excision theorem. Some computations of higher homotopy groups. Freudenthal's suspension theorem and applications.
Keywords
topological space, fundamental group, covering space, functor, homotopy, homotopy groups, CW-complex
Educational Material Types
- Notes
- Book
Course Organization
| Activities | Workload | ECTS | Individual | Teamwork | Erasmus |
|---|---|---|---|---|---|
| Lectures | 39 | ||||
| Seminars | 20 | ||||
| Reading Assigment | 183 | ||||
| Project | 35 | ||||
| Written assigments | 20 | ||||
| Exams | 3 | ||||
| Total | 300 |
Student Assessment
Description
1. Written or oral exam (50-70%).
2. Short presentation of a topic and written report (30-50%).
Student Assessment methods
- Written Exam with Extended Answer Questions (Summative)
- Written Assignment (Formative)
- Oral Exams (Summative)
- Performance / Staging (Formative, Summative)
- Written Exam with Problem Solving (Summative)
- Report (Formative, Summative)
Bibliography
Additional bibliography for study
1. G. E. Bredon "Topology and Geometry"
2. A. Hatcher "Algebraic Topology"
3. J. P. May "A Concise Course in Algebraic Topology"
4. R. M. Switzer "Algebraic Topology - Homology and Homotopy"
5. T. tom Dieck "Algebraic Topology"
6. J. Munkres “Topology”
Last Update
11-05-2025
