About the subject

Aim: Learn conceptual and computations aspects of characteristic classes (Stiefel—Whitney, Euler, Chern and Pontrjagin classes) and their implications to manifold topology.

Description: Characteristic classes are cohomology classes associated to vector bundles over manifolds. These classes often have geometrical interpretation with corresponding geometrical/topological implications. For example, Stiefel—Whitney classes of a bundle measure orientability, possibility of introducing Spin structures on the bundle and how many different such structure one can introduce. Also, the Euler class of a rank n vector bundle over an n-dimensional manifold counts the number of zeros of sections of said bundle.

In this course we will develop the theory of characteristic classes and computation methods (Schubert calculus, Chern-Weil theory). We will study relations between different characteristics classes, obstruction problems and multiplicative sequences.

Characteristic classes

Thursday, June 21, 2012 (Joost-Joost-Jules)

Appendix C: actual computations on a smooth manifold.

Thursday, June 14, 2012 (Joost-Joost-Jules)

Appendix C: actual computations on a smooth manifold.

Thursday, June 7, 2012 (Reinier and Ralph)

Characteristic numbers and the cobordism ring. Chapters 16 and 17.

Thursday, May 31, 2012 (Reinier and Ralph)

Characteristic numbers and the cobordism ring. Chapters 16 and 17.

Thursday, May 24, 2012 (Simen, Julian, Joey and Bram)

Simen and Julian will finish the chapter on obstruction theory and Joey and Bram will do the chapter on Pontrjagin classes Chapters 12 and 15.

Thursday, May 10, 2012 (Simen, Julian and Bram)

Bram will finish the chapter on Chern classes (Chapter 13) and Simen and Julian will talk about obstruction theory. Chapter 12.

Thursday, May 3, 2012 (Joey and Bram)

Joey and Bram will finish today the chapter on Chern classes. Chapter 13

Thursday, April 26, 2012 (Yihang and Joey)

Today Yihang finished the section on computations on a smooth manifold by proving Poincare duality and Wu’s formula. In the second hlaf of the lecture, Joey defined complex vector bundles and defined Chern classes of complex vector bundles inductively. The first few properties of Chern classes were proved. This is covered in chapters 11, 13 and 14.

Thursday, April 19, 2012 (Dali and Yihang)

Dali and Yihang continued with the chapter computations on smooth manifolds.

Thursday, April 12, 2012

Lecture cancelled because of the Dutch Mathematics Congress.

Thursday, April 5, 2012 (Joost Nuiten and Dali)

Joost finished his presentation on the Thom isomorphism and Dali started with chapter 11: Computations on a smooth manifold.

Thursday, March 29, 2012 (Joost Nuiten)

This lecture Joost proved the Thom isomorphism theorem in a very general form (over a paracompact base and with coefficients in an arbitrary ring). This is covered in Chapter 10.

Thursday, March 22, 2012 (Jules)

This lecture Jules introduced the Euler class of an oriented bundle and went on to prove its main properties. One of the main ingredients in the definition were the Thom class and corresponding Thom isomorphim. This material is covered in Chapter 9.

Thursday, March 15, 2012 (Reinier)

This lecture Reinier explained the (co)homology of Grasmanians and went on to prove existence and uniqueness of Stiefel--Whitney classes. This is covered in Chapters 7 and 8.

Thursday, March 8, 2012 (Ralph)

This lecture Ralph explained the cell structure of Grasmannians. This is treated in Chapter 6.

Thursday, March 1, 2012 (Joost & Looijenga)

This lecture we saw that any rank n bundle over a paracompact space can be realized as the pull back of the tautological bundle over Gn and that any two bundles realized in such fashion are bundle homotopic. We finished Chapter 5.

Thursday, February 23, 2012 (Robin continued; Joost & Looijenga)

This lecture Robin finished Chapter 4: he introduced Stiefel--Whitney numbers, then we saw these numbers are  cobordism invariant and in fact they fully determine the cobordism class of a space.

Then Joost started our study of Grasmanians (Chapter 5): we saw the definitions of Grasmanians and  Stiefel manifolds, tautological bundle  and saw eventually that given a bundle over a compact space K, one can realize it as a pull back of the tautological bundle by an appropraite map K ---> Gn(n+k) for K large enough.

Thursday, February 16, 2012 (Robin & Cavalcanti)

We started the lecture with the proof that the tautological bundle over RPn is not trivial. Then we introduced the axioms for Stiefel--Witney classes and obtained simple consequences. We finished the lecture computing the total Stiefel--Whitey class of the tangent space of RPn.

Thursday, February 9, 2012 (Cavalcanti)

Review of vector bundles over topological and smooth manifolds: definitions and constructions (Chapters 1, 2 & 3)


Practical information

The book we will be using as reference for this course is Milnor and Stasheff’s “Characteristic classes”.

The lectures will take place in room 611 of the Mathematics building on Thursdays from 11:00 to 13:00, starting on the 9th of February.

The course will be run in seminar style and the final mark will be determined by the oral presentations plus discussions (85%), and by the general involvement in the discussions that follow the presentations of fellow students which may include, among other things, handing in homework assignments (15%).