### 2016　Topology

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Instructor(s)
Endo Hisaaki
Class Format
Lecture
Media-enhanced courses
Day/Period(Room No.)
Tue3-4(H136)  Fri3-4(H136)
Group
-
Course number
MTH.B341
Credits
2
2016
Offered quarter
4Q
Syllabus updated
2016/4/27
Lecture notes updated
-
Language used
Japanese
Access Index ### Course description and aims

The main subject of this course is basic concepts of homology groups and fundamental groups. After introducing some notions for homotopy and deformation retraction, we explain basic notions for simplicial complexes, such as simplex, simplicial complex, simplicial maps, barycentric subdivision, and simplicial approximation. We next 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.
The homology group and the fundamental group are the most fundamental notions in topology, and are prototypes of topological invariants. In this course students will have a chance to know the concepts of "invariants" and their "functoriality" in studying the homology group and the fundamental group.

### 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 group of simple topological spaces

### Keywords

homotopy, deformation retract, simplicial complex, simplicial map, simplicial approximation, chain group, boundary homomorphism, homology group, induced homomorphism, Euler number, the Mayer-Vietoris exact sequence, homotopy invariance, loop, 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 equivalent, deformation retract, contractible, 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 number 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 the Euler-Poicare formula, cone, acyclic, calculation of homology groups Details will be provided during each class session
Class 9 chain map, induced homomorphism, functoriality Details will be provided during each class session
Class 10 connecting homomorphism, the Mayer-Vietoris exact sequence Details will be provided during each class session
Class 11 product complex, chain homotopy, homotopy invariance (1) Details will be provided during each class session
Class 12 homotopy invariance (2), applications of homology groups Details will be provided during each class session
Class 13 path, loop, product, inverse, fundamental group Details will be provided during each class session
Class 14 induced homomorphism, change of base points, homotopy invariance Details will be provided during each class session
Class 15 free product of groups, the Seifert-van Kampen theorem Details will be provided during each class session

None required

### Reference books, course materials, etc.

Allen Hatcher, Algebraic Topology, Cambridge University Press

### Assessment criteria and methods

final examination (70%), exercises (30%)

### 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). 