### 2020　Topology

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Instructor(s)
Endo Hisaaki
Class Format
Lecture    (ZOOM)
Media-enhanced courses
Day/Period(Room No.)
Tue5-6(H111)  Fri5-6(H111)
Group
-
Course number
MTH.B341
Credits
2
2020
Offered quarter
4Q
Syllabus updated
2020/9/18
Lecture notes updated
-
Language used
Japanese
Access Index

### Course description and aims

The main goal of this course is to cover basic concepts of homology groups and fundamental groups. The homology group and the fundamental group are fundamental notions in topology, and are prototypes of topological invariants. After introducing the notions of homotopy, we explain some basics for simplicial complexes, such simplicial maps, barycentric subdivision, and simplicial approximation. Next, we introduce the chain group and the homology group of a simplicial complex and the induced map of a simplicial map, and prove the homotopy invariance of the homology group. We finally define the fundamental group of a topological space and show the Seifert-van Kampen theorem.

### Student learning outcomes

Students are expected to:
- Be able to determine whether a given set of simplices is a simplicial complex
- Understand the precise statement and importance of the simplicial approximation theorem
- Be able to calculate the homology group of a given simplicial complex
- Be able to calculate the fundamental groups of simple topological spaces

### Keywords

homotopy, deformation retract, simplicial complex, simplicial map, chain group, boundary homomorphism, homology group, induced homomorphism, Euler number, the Mayer-Vietoris exact sequence, homotopy invariance, fundamental group, the Seifert-van Kampen theorem

### Competencies that will be developed

 ✔ Specialist skills Intercultural skills Communication skills Critical thinking skills Practical and/or problem-solving skills

### Class flow

Standard lecture course

### Course schedule/Required learning

Course schedule Required learning
Class 1 overview, product space, quotient space, homotopy Details will be provided during each class session
Class 2 homotopy equivalence, deformation retract, contractibility, simplex, face, barycentric coordinate Details will be provided during each class session
Class 3 simplicial complex, subcomplex, polyhedron, simplicial decomposition, abstract simplicial complex Details will be provided during each class session
Class 4 geometric realization, simplicial map, isomorphism, barycenter, joinable, join Details will be provided during each class session
Class 5 barycentric subdivision, open star, simplicial approximation, Lebesgue's lemma Details will be provided during each class session
Class 6 simplicial approximation theorem, orientation, chain group, boundary homomorphism Details will be provided during each class session
Class 7 cycle, boundary cycle, homology group, Betti number, Euler characteristic Details will be provided during each class session
Class 8 Euler-Poincare formula, cone complex, acyclic Details will be provided during each class session
Class 9 calculation of homology groups Details will be provided during each class session
Class 10 chain map, induced homomorphism, functoriality Details will be provided during each class session
Class 11 connecting homomorphism, Mayer-Vietoris exact sequence Details will be provided during each class session
Class 12 product complex, chain homotopy Details will be provided during each class session
Class 13 homotopy invariance of homology groups Details will be provided during each class session
Class 14 Applications of homology groups Details will be provided during each class session

### Out-of-Class Study Time (Preparation and Review)

To enhance effective learning, students are encouraged to spend approximately 100 minutes preparing for class and another 100 minutes reviewing class content afterwards (including assignments) for each class.
They should do so by referring to textbooks and other course material.

None required

### Reference books, course materials, etc.

Allen Hatcher, Algebraic Topology, Cambridge University Press

### Assessment criteria and methods

examination (50%), assignment (50%)

### Related courses

• MTH.B301 ： Geometry I
• MTH.B302 ： Geometry II

### Prerequisites (i.e., required knowledge, skills, courses, etc.)

Students are expected to have passed Introduction to Topology I (MTH.B201), Introduction to Topology II (MTH.B202), Introduction to Topology III (MTH.B203), Introduction to Topology IV (MTH.B204), Introduction to Algebra I (MTH.A201), Introduction to Algebra II (MTH.A202), Introduction to Algebra III (MTH.A203), Introduction to Algebra IV (MTH.A204).