The aim of this lecture course is to familiarize students with the basic language of, and some fundamental theorems in differential topology. This course will be succeeded by [MTH.B404 : Advanced topics in Geometry D].
As an outcome, students are expected to
・understand the notions of homotopy group, cobordism ring, the degree of a map etc.
・be familiar with the classification of surfaces, the method of smooth approximation, general position arguments, the chain complex for the homology group of a CW complex etc.
vector field, rotation, genus, homotopy group, degree, immersion, cobordism, transversality
|✔ Specialist skills||Intercultural skills||Communication skills||Critical thinking skills||Practical and/or problem-solving skills|
Standard lecture course
|Course schedule||Required learning|
|Class 1||vector fields and their rotation (Whitney index)||Details will be provided during each class session|
|Class 2||classification of closed surfaces||Details will be provided during each class session|
|Class 3||smooth approximation of continuous maps||Details will be provided during each class session|
|Class 4||homotopy groups||Details will be provided during each class session|
|Class 5||immersions, submersions, transversality||Details will be provided during each class session|
|Class 6||degree of a map||Details will be provided during each class session|
|Class 7||homology groups of CW complexes||Details will be provided during each class session|
|Class 8||cobordism rings, Pontryagin--Thom construction||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.
J. Milnor: Topology from the differentiable viewpoint
W. Fulton: Algebraic topology
Evaluation will be based on exams and homework. Details will be provided during class sessions.
Students are expected to have passed [Geometry I], [Geometry II] and [Geometry III].
to be determined