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.

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 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

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