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| MATL7950 | Algebraic Topology | 3+0+0 | ECTS:7.5 | | Year / Semester | Spring Semester | | Level of Course | Third Cycle | | Status | Elective | | Department | DEPARTMENT of MATHEMATICS | | Prerequisites and co-requisites | None | | Mode of Delivery | | | Contact Hours | 14 weeks - 3 hours of lectures per week | | Lecturer | Doç. Dr. Tane VERGİLİ | | Co-Lecturer | None | | Language of instruction | | | Professional practise ( internship ) | None | | | | The aim of the course: | | The goal of this course is to introduce the fundamental and crucial algebraic structures in algebraic topology such as fundamental group, homotopy group and homology groups. Therefore, students gain the ability to compute these algrebraic structures related to topological spaces. |
| Programme Outcomes | CTPO | TOA | | Upon successful completion of the course, the students will be able to : | | | | PO - 1 : | learn the basic algebraic methods in topology. | | 1,3, | | PO - 2 : | discuss a topological problem with algebraic methods. | | 1,3, | | PO - 3 : | compute the homology and homotopy groups of topological spaces. | | 1,3, | | CTPO : Contribution to programme outcomes, TOA :Type of assessment (1: written exam, 2: Oral exam, 3: Homework assignment, 4: Laboratory exercise/exam, 5: Seminar / presentation, 6: Term paper), PO : Learning Outcome | | |
| Categories and functors, (path) homotopy and fundamental groups, covering spaces, homotopy equivalence, homotopy groups, simplicial complex, simplicial homology, singular homology, Eilenberg-Steenrod Axioms, Mayer-Vietoris, Excision, Künneth Formula and universal coefficient theorem |
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| Course Syllabus | | Week | Subject | Related Notes / Files | | Week 1 | Continuity and paths in a space | | | Week 2 | Homotopy and path homotopy | | | Week 3 | Fundamental group construction | | | Week 4 | Covering spaces | | | Week 5 | The fundamental group of a circle | | | Week 6 | | | | Week 7 | Categories and functors | | | Week 8 | Simplex, simplicial complex, triangulation | | | Week 9 | Midterm Examination | | | Week 10 | Short and long exact sequences | | | Week 11 | Simplicial Homology | | | Week 12 | Singular homology | | | Week 13 | Mayer-Vietoris ve Excision Teoremi | | | Week 14 | Ext and Tor functors | | | Week 15 | Universal Coefficient Theorem in Homology and Künneth Formula | | | Week 16 | Final Examination | | | |
| 1 | Rotman Joseph J. An Introduction to Algebraic Topology, Springer, ISBN: 978-1-4612-4576-6. | | | |
| 1 | Hatcher Allan, Algebraic Topology, https://pi.math.cornell.edu/~hatcher/AT/AT.pdf | | | 2 | Spainer Edwin H. 1966; Algebraic Topology, Springer-Verlag, ISBN-10: 0387944265 | | | 3 | Massey William S., Algebraic Topology: An Introduction, Springer ISBN-10: 0387902716 | | | |
| Method of Assessment | | Type of assessment | Week No | Date | Duration (hours) | Weight (%) | | Mid-term exam | 9 | - | 2 | 30 | | In-term studies (second mid-term exam) | 12 | - | 2 | 20 | | End-of-term exam | 16 | - | 2 | 50 | | |
| Student Work Load and its Distribution | | Type of work | Duration (hours pw) | No of weeks / Number of activity | Hours in total per term | | Yüz yüze eğitim | 3 | 14 | 42 | | Sınıf dışı çalışma | 7 | 14 | 98 | | Arasınav için hazırlık | 8 | 2 | 16 | | Arasınav | 3 | 2 | 6 | | Ödev | 5 | 5 | 25 | | Dönem sonu sınavı için hazırlık | 10 | 2 | 20 | | Dönem sonu sınavı | 3 | 1 | 3 | | Total work load | | | 210 |
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