MATH710 LOW DIMENSIONAL TOPOLOGY
| Course Code: | 2360710 |
| 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: | Assoc.Prof.Dr. MEHMET FIRAT ARIKAN |
| Offered Semester: | Fall Semesters. |
Course Objectives
This course provides students:
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Understand and classify low-dimensional manifolds
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Develop fluency with handlebody theory
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Apply Kirby calculus techniques
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Explore 3-manifold topology as the boundary theory of 4-manifolds
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Understand the role of smooth structures in 4 dimensions
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Develop skills in diagrammatic and combinatorial reasoning
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Use the techniques of the course to approach open problems in 4-manifold topology.
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Build foundational understanding for advanced topics such as gauge theory, Seiberg–Witten invariants, or symplectic topology.
Course Content
Courses not listed in catalogue. Contents vary from year to year according to interest of students and instructor in charge. Typical contents include contemporary developments in Algebra, Analysis, Geometry, Topology, Applied Mathematics.
Course Learning Outcomes
By the end of this course, students will be able to:
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Break down 3- and 4-manifolds into handles of various indices.
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Determine how handle attachments affect topology and smooth structure.
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Translate between framed link diagrams and handlebody decompositions of 4-manifolds.
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Accurately apply handle slides, blow-ups, and cancellations in diagrammatic form.
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Perform Kirby moves to show that two different diagrams represent diffeomorphic 4-manifolds.
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Simplify complicated handlebody representations using standard calculus moves.
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Calculate homology and fundamental groups of 3- and 4-manifolds using handle decompositions and Mayer–Vietoris sequences.
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Determine intersection forms and Euler characteristics from handle structures.
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Represent closed, oriented 3-manifolds as surgery on framed links.
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Use Lickorish–Wallace theorem and basic surgery operations to build 3-manifolds.
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Explain the uniqueness of topological structures and the abundance of exotic smooth structures on certain 4-manifolds.
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Analyze how 3-manifolds arise as boundaries of 4-manifolds.
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Understand cobordisms and their role in comparing different smooth structures.
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Prepare for research involving gauge theory, symplectic topology, or smooth manifold classification.
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