MATH538 ALGEBRAIC TOPOLOGY II
| Course Code: | 2360538 |
| METU Credit (Theoretical-Laboratory hours/week): | 3 (3.00 - 0.00) |
| ECTS Credit: | 8.0 |
| Department: | Mathematics |
| Language of Instruction: | English |
| Level of Study: | Graduate |
| Course Coordinator: | Prof.Dr. YILDIRAY OZAN |
| Offered Semester: | Fall Semesters. |
Course Objectives
By the end of the course the student will learn
- the the definion and basic properties of singular cohomology groups of the the spaces,
- the definition and basic properties of singular cohomology ring of spaces,
- the homological definition of orientation on manifolds and Poincaré Duality,
- the definition and basic properties of homotopy groups.
Course Content
Comohology groups, Universal Coefficient Theorem, comohology of spaces. Products in comohology, Kunneth formula. Poincare duality. Universal coefficient theorem for homology. Homotopy groups.
Course Learning Outcomes
At the end of the course the student will learn
- the definion of singular cohomology groups and relative cohomology groups of a space,
- Universal Coefficient Theorem for homology and cohomology,
- long exact sequence, homotopy invariance, excision for singular cohomology groups,
- cellular cohomology,
- the definition and basic properties of cup product and coholomogy ring of a space,
- computation of cohomology rings of real projective space and some familiar spaces,
- the homological definion of orientation on (topological) manifolds, fundamental class,
- the definition of cap product; Poincaré Duality,
- the definition of homotopy groups and basic constructions,
- the definition and basic properties of relative homotopy groups and homotopy exact sequence of a pair.
Program Outcomes Matrix
| Level of Contribution | |||||
| # | Program Outcomes | 0 | 1 | 2 | 3 |
| 1 | Acquires mathematical thinking skills (problem solving, generating ways of thinking, forming correspondence, generalizing etc.) and can use them in related fields. | ✔ | |||
| 2 | Gains academic maturity through self-study. | ✔ | |||
| 3 | Can design mathematics related problems, devise solution methods and apply them when appropriate. | ✔ | |||
| 4 | Carries out parts of a mathematical research program independently. | ✔ | |||
| 5 | Has a command of Turkish and English languages so that he/she can actively communicate (read, write, listen and speak). | ✔ | |||
| 6 | Contributes to solving global, environmental and social problems either individually or as being part of a social group. | ✔ | |||
| 7 | Respects ethical values and rules; applies them in professional and social issues. | ✔ | |||
| 8 | Can work cooperatively in a team and also individually. | ✔ | |||
| 9 | Gets exposed to academic culture through interaction with others. | ✔ | |||
| 10 | Comprehends necessity of knowledge, can define it and acquires it; uses knowledge effectively and shares it with others. | ✔ | |||
0: No Contribution 1: Little Contribution 2: Partial Contribution 3: Full Contribution