Earlier course notes

 Homological Algebra

This course is an introduction to basic concepts in homological algebra; intended for students with just a basic background in algebra. Homological algebra studies certain sequences of abelian groups called chain complexes. An typical example is the complex of linear combinations of simplices in a topological space. To such complexes one may assign an invariant called their homology, which gives the subject its name. Homological algebra provides tools for studying homology generally and by explicit computation, not only in topology but also homology of groups, of Lie algebras, and of various structures in algebraic geometry.

 

 Algebraic Topology

The course will start with a reminder about the fundamental group which most students will have seen, and the homotopy relation on maps. Next, we will define the higher homotopy groups and prove some basic properties about them. We will discuss the action of the fundamental group on these groups and and the Serre long exact sequence of a fibration. This will enable us to compute some elementary examples.

We will discuss CW-complexes and a proof of Whitehead's theorem about the construction of maps from the information about homotopy groups. Further, we will use CW-complexes to construct the Postnikov tower of a space, a context in which the famous Eilenberg-Mac Lane spaces come up. We also hope to discuss the Freudenthal suspension theorem, a key result that lies at the basis of stable homotopy theory.

 

 Homology

Singular homology groups are algebraic invariants of spaces: for every space there are such groups and every map of spaces induces a map between the corresponding groups. These invariants turn out to be rather computable, and they allow for some immediate geometric applications.

In this course, we will establish some key properties of these homology groups like the homotopy-invariance and the excision theorem. A convenient variant is provided by singular homology with coefficients - a framework which makes necessary a short discussion of basic homological algebra including tensor and torsion products. For CW complexes, there is also the more combinatorial cellular homology theory. The course culminates in a proof that singular homology and cellular homology agree on CW complexes. This allows for more explicit calculations in examples of interest (e.g., projective spaces).

 

 Cohomology

The basic philosophy of algebraic topology consists of assigning algebraic invariants to topological spaces. These invariants are expected to be interesting enough to capture important geometric information and, at the same time, to be accessible to actual calculations. A prototype is given by singular homology theory in which case the invariants are abelian groups. In this course, we will study closely related invariants, namely singular homology with coefficients and, most of the time, singular cohomology. It turns out that many of the key formal properties of singular homology (homotopy invariance, excision, and so on) are also enjoyed by these variants. A key difference occurs when one considers cohomology with coefficents in a commutative ring; in that case, the singular cohomology groups of a space can be turned into a graded-commutative ring, and this additional product structure is very useful in applications.

 

 Topological K-Theory

Topological K-theory, the first generalized cohomology theory to be studied thoroughly, was introduced in a 1961 paper by Atiyah and Hirzebruch, where they adapted the work of Grothendieck on algebraic varieties to a topological setting. Since that time, topological K-theory has become a powerful and indispensable tool in topology, differential geometry, and index theory.

In this course we will study the theory of vector bundles and related algebraic notions, followed by the definition of K-theory and proofs of some important theorems in the subject, such as the Bott periodicity theorem.

 

 Representations of Finite Groups

These are the notes of a short 8 lecture course on the theory of representations of finite groups, taught several years to second year undergraduate students in Nijmegen. The contents of the course are the basics of the theory of representations of finite groups, in particular the correspondence between representations and characters, the representation theory of the symmetric group, and induced representations. Since the topic involves a lot of linear algebra, the notes include an appendix that summarizes some important concepts from linear algebra. Furthermore, a section on category theory is included, to put the theory of representations in a broader perspective.