Algebraic Topology II - M2 - 8EC

  1. Prerequisites: 

A working knowledge of algebraic topology at the level of Algebraic Topology I: This includes, but is not limited to: (co)homology, Eilenberg-Steenrod axioms, CW-complexes, cellular (co)homology, representability of cohomology by Eilenberg-Maclane spaces, degree theory, fundamental group, definition of higher homotopy groups. 

Manifold topology: (Smooth) manifolds, smooth maps, tangent bundle, derivative, embedding, immersion, submersion, approximation of continuous maps by smooth maps up to homotopy, inverse function theorem, implicit function theorem. 

 

  1. Aim of the course

 

The content of algebraic topology varies year by year. This year the focus lies on fiber bundle theory. 

 

Fiber bundles are twisted products of spaces. Many interesting topological spaces arise naturally as fiber bundles: for example the tangent bundle of a manifold is a fiber bundle, but projective spaces, and more generally grassmannian manifolds are fiber bundles. In this course we will study fiber bundles from an algebraic topological perspective. We intent to cover the following topics:

  • Cups products and Poincare duality. 

  • Definition of fiber bundle, principal G-bundle

  • Classification of bundles by homotopy classes of maps into classifying spaces

  • The cohomology rings of Grassmannian manifolds. 

  • Characteristic classes of fiber bundles

  • The splitting principle 

  • The Thom isomorphism

  • An introduction to cobordism theory and the relation with classifying spaces

 

 

  1. Rules about homework / Exam

There are two homework problem sets that will be graded with grade H_1 and H_2. There is one final written exam, with grade F. The weighted average A is calculated by

A=H_1/10+H_2/10+8 F/10

To pass the course the grades should satisfy F>=5, and A>= 5.5. 

 

If F>=5, the grade of the course is A

If F<5 the grade of the course is F

 

There is one resit of the whole course. For the resit the homework does not count. 

 

  1. Lecture notes / literature

There will be lecture notes made available. 

We will sometimes refer to the following additional literature

 

Husemöller: Fiber bundles

Milnor and Stasheff: Characteristic classes. 

Hatcher: Algebraic Topology

Hatcher: Vector bundles and K-Theory

 

  1. Lecturers

Renee Hoekzema (VU)

Thomas Rot (VU)