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School of Environment and Society
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- Common courses
- Liberal arts and basic science courses
- Academic unit or major
- Graduate major in Mathematics
- Instructor(s)
- Nosaka Takefumi
- Class Format
- Lecture
- Media-enhanced courses
- Day/Period(Room No.)
- Fri5-6(H104)
- Group
- -
- Course number
- MTH.B501
- Credits
- 1
- Academic year
- 2018
- Offered quarter
- 1Q
- Syllabus updated
- 2018/3/20
- Lecture notes updated
- -
- Language used
- Japanese
- Access Index
-
Course description and aims
In this course, we mainly focus on studies of manifolds with betti number >0.
In particular, we study such invariants, including Alexander polynomial, Reidemeister torsion and Blanchfield pairing. This course is an introduction to such invariants.
Student learning outcomes
Introduction to Alexander polynomial;
(1) In low-dimensional case, we study diagrammatic computation of Alexander polynomials.
(2) We study Blanchfield duality and pairings of infinite cyclic coverings.
(3) We look at applications of Reidemeister torsion.
Keywords
Fundamental group, covering spaces, homology with local coefficients, Alexander polynomial, duality theorem, Reidemeister torsion
Competencies that will be developed
| ✔ Specialist skills | Intercultural skills | Communication skills | Critical thinking skills | ✔ Practical and/or problem-solving skills |
Class flow
Standard lecture. Homeworks will be assigned in some classes.
Course schedule/Required learning
| Course schedule | Required learning | |
|---|---|---|
| Class 1 | Review; fundamental group and coverings | Details will be provided during each class session |
| Class 2 | Alexander polynomial and Seifert surfaces | Details will be provided during each class session |
| Class 3 | Homology with local coefficients and Fox derivation | Details will be provided during each class session |
| Class 4 | Skein relations and computation of Alexander polynomial | Details will be provided during each class session |
| Class 5 | Milnor dualiy of infinite cyclic coverings | Details will be provided during each class session |
| Class 6 | Blanchfield pairing | Details will be provided during each class session |
| Class 7 | Reidemeister torsion I;definition | Details will be provided during each class session |
| Class 8 | Reidemeister torsion II; application | Details will be provided during each class session |
Textbook(s)
None required
Reference books, course materials, etc.
Vladimir Turaev, Introduction to Combinatorial Torsions, Lectures in Mathematics. ETH Zürich
W.B.Raymond Lickorish , An Introduction to Knot Theory, Graduate Texts in Mathematics
Assessment criteria and methods
Graded by homeworks
Related courses
- MTH.B341 ： Topology
- MTH.B508 ： Advanced topics in Geometry H1
- MTH.B408 ： Advanced topics in Geometry D1
- MTH.B211 ： Introduction to Geometry I
Prerequisites (i.e., required knowledge, skills, courses, etc.)
Basics of manifolds, topology, and homology theory
Contact information (e-mail and phone) Notice : Please replace from "[at]" to "@"(half-width character).
nosaka[at]math.titech.ac.jp
Office hours
N/A.
Contact by E-mails, or at the classroom.
Other
Not in particular
