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.
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
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
✔ Specialist skills | Intercultural skills | Communication skills | Critical thinking skills | Practical and/or problem-solving skills |
Standard lecture course
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 |
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
Allen Hatcher, Algebraic Topology, Cambridge University Press
examination (50%), assignment (50%)
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).