Reading seminar in Algebraic Topology

The goal of this seminar is to study one of the three topics described below: Vector bundles and K-Theory, Spectral sequences and Poincare duality for manifolds. The choice depends on the interest of the students and on their background. The focus is on concrete examples rather than on abstract exposition.

Contacts: Dietrich Notbohm <notbohm (at) few.vu.nl> or Alvise Trevisan <A.Trevisan (at) few.vu.nl>.

News
10 Sep 2008 The topic chosen for the reading seminar is "Vector bundles and K-Theory".
05 Sep 2008 The first meeting will be Wednesday September 10 in room R5.26 at 16.15h, where we will choose the topics to be discussed during the semester and fix a day and time for the meetings.

Place and time We meet every week at 16.15 on the second floor, R wing and then move to any free froom..

Schedule of meetings

Monday, 15/09/2008, 16:15 - 17:15
Speaker: Alvise Trevisan
Title: Introduction to vector bundles.
References: [HVB] 1.1 and [MCC] §2 and §3

Monday, 22/09/2008, 16:15 - 17:15
Speaker: Jacobien Carstens
Title: Classifying vector bundles, pt. I. Vector bundles over spheres.
References: [HVB] 1.2 and [MCC] §5

Monday, 29/09/2008, 16:15 - 17:15
Speaker: Jacobien Carstens and Maurits van der Meer
Title: Classifying vector bundles, pt. II. Clutching functions. The universal bundle.
References: [HVB] 1.2 and [MCC] §5

Monday, 06/10/2008, 16:15 - 17:15
Speaker: Maurits van der Meer
Title: Classifying vector bundles, pt. III. A cell structure for grassmannians.
References: [HVB] 1.2 and [MCC] §5

Monday, 13/10/2008, 16:15 - 17:15
Speaker: Alvise Trevisan
Title: Introduction to K-Theory.
References: [HVB] 2.1 and [AKT] §2.1 and §2.2

The two prototypal examples of VECTOR BUNDLES are the Möbius band, a "twisted" product of a line and a circle, and the annulus, an actual product. Both of them can be thought of as a continuously varying collection of lines.

Vector bundles are generalizations of these two objects, with an arbitrary topological space in place of the circle (the "base space") and an arbitrary (finite dimensional) vector space in place of the line (the "fiber"). Vector bundles can be invested with some algebraic structure: there is a direct sum operation which, on the fibers, reduces to the familiar direct sum of vector spaces. K-THEORY uses this structure to define algebraic invariants of the spaces in question, in some ways more powerful than ordinary cohomology.

Some very deep results obtained using this machinery are, for example, the theorem that there are no finite dimensional division algebras over $ \mathbb{R}$ in dimensions other than 1,2,4 and 8 (corresponding respectively to real numbers, complex numbers, quaternions and Cayley octonions). Another such example is the theorem stating that the only spheres whose tangent bundles are product bundles are $ S^1, S^3$ and $ S^7$ .

To introduce vector bundles and study their basic properties only a prior knowledge of linear (and abstract) algebra and point-set topology is required, even though a good command of algebraic topology would provide both motivation and more insight to the matter.

Possible sub-topics include

Literature:

  1. [HVB] Hatcher, A., Vector bundles and K-Theory, available online here.
  2. [MCC] Milnor, J.W. and Stasheff, J.D., Characteristic classes.
  3. [AKT] Atiyah, M., K-Theory.

SPECTRAL SEQUENCES were introduced in the 1940's by Jean Leray as a mean to compute sheaf cohomology, but have been generalized and extended to a large number of other situations. The idea is to start from a collection of chain complexes, fitting together in a suitable way, and pass to their cohomology. The resulting data will form agaian a collection of chain complexes, so we can iterate the procedure. If the starting data is "nice", the procedure will eventually lead to the wanted object (e.g. the cohomology of a fibration, Tor and Ext groups of modules, algebraic K-theory of a field, algebraic de Rham cohomology of a variety, etc.).

Possible examples of spectral sequences to be discussed include

Literature:

  1. Hatcher, A., Spectral sequences in algebraic topology, available online here.
  2. McCleary, J., A user's guide to spectral sequences.

POINCARE DUALITY is a fundamental result in algebraic topology, relating the cohomology and homology groups of a (topological) manifold. The statement is that if $ M$ is an $ n$ -dimensional compact oriented manifold, the $ k$ -th cohomology group $ H^k(M)$ is isomorphic to the $ n-k$ -th homology group $ H_{n-k}(M)$ . In particular, all cohomology groups vanish in dimension greater than $ n$ .

Possible topics to be discussed include:

Literature:

  1. Bredon, G.E., Geometry and topology.