# Course on Cohomology Theory (second semester 2012/2013)

This is a course on basic aspects of (singular) cohomology theory offered by Moritz Groth. The course consists of ten lectures.

Time and place: Monday (first session: February 4), 13.00-16.00, room: HG02.702
Prerequisites: Basic singular homology theory (see, for example, here) and elementary notions of homotopy theory.
Examination: Grades (6EC) will be given based on an oral examination and on an optional talk in the block seminar.

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 the closely related theory of singular cohomology (in its many variants: for pairs of spaces, reduced, with coefficients). It turns out that singular cohomology with coefficients in a commutative ring carries more algebraic structure. This is witnessed by the existence of certain product structures turning the singular cohomology of a space into a graded-commutative ring.

Singular (co)homology of finite CW complexes has certain key formal properties, and it can be shown that they already determine the theory (which establishes the reassuring fact that various approaches to a (co)homology theory for spaces coincide - at least on nice spaces). We begin by a short recap of homology theory which basically amounts to establishing these key formal properties. It is then rather easy to extend these results to homology with coefficients. Next, we discuss basic aspects of singular cohomology, including a universal coefficients result. As a preparation for the study of product structures in cohomology, we give a careful treatment of Eilenberg-Zilber equivalences and the Künneth theorem. A few applications of product structures will be mentioned, and we also present some basic explicit examples. The final lecture will indicate that in a certain respect cohomology theory can be considered as a part of homotopy theory.

Lecture notes (updated course notes and exercise sheets can be found here):
Lecture 01   Recap of basics in singular homology theory
Lecture 02   Homotopy invariance of singular homology theory
Lecture 03   More fundamental theorems in singular homology theory
Lecture 04   Singular homology with coefficients
Lecture 05   Basics in singular cohomology
Lecture 06   Universal coefficient theorem in singular cohomology
Lecture 07   Eilenberg-Zilber equivalences and the Künneth theorems
Lecture 08   Product structures in cohomology
Lecture 09   Some appplications of multiplicative structures
Lecture 10   Cofibers and (co)homology

References: References for such a course include the following ones:
Davis, Kirk;  Lecture notes in algebraic topology
Eilenberg, Steenrod;  Foundations of algebraic topology
Hatcher;  Algebraic Topology
Spanier;  Algebraic Topology
Switzer;  Algebraic Topology: Homotopy and Homology