Course on Algebraic Topology II (second semester 2013/2014)

This is a course on some aspects of algebraic topology offered by Moritz Groth (first six lectures) and Ieke Moerdijk (second six lectures).

Time and place: Thursday (first session: February 20), 13.30-15.30, room: HG03.085 (Hilbert space)
Exercice sessions: with Giovanni Caviglia, Friday, 13.30-14.30, room: HG03.085 (Hilbert space).
Prerequisites: Some background in algebraic topology (for example the content of the course Algebraic Topology I).
Examination: Grade based on presentations by students and oral exam. See below for the dates for both the presentations and the orals. The presentations will take place in HG00.310.

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.

We begin by a short discussion of singular homology with coefficients, including a digression on tensor and torsion products. Next, we discuss basic aspects of singular cohomology, including a universal coefficients result which indicates a relation to homology. The Künneth theorem allows us to calculate the homology of a product space and serves us as a preparation for the study of product structures in cohomology. A few applications of this product structure are mentioned, and we also present some explicit examples. We conclude this course by an introduction to cohomology operations, yielding an even more refined algebraic structure on the cohomology of a space.

Lecture notes: (thanks to Matija Bašić for typing sections 7-12)
Section 01   Singular homology with coefficients
Section 02   Universal coefficient theorem in singular homology
Section 03   Basics in singular cohomology
Section 04   Universal coefficient theorem in singular cohomology
Section 05   Eilenberg-Zilber equivalences and the Künneth theorems
Section 06   Product structures in cohomology
Section 07   Presheaves and Čech cohomology
Section 08   Comparison of Čech and singular cohomology
Section 09   Fiber bundles and principal bundles
Section 10   Induced bundles and vector bundles
Section 11   Homotopy invariance of fibre bundles

Exercise sheets: (thanks to Giovanni Caviglia for creating and typing exercise sheets 7-12)
Exercise sheet 01
Exercise sheet 02
Exercise sheet 03
Exercise sheet 04
Exercise sheet 05
Exercise sheet 06
Exercise sheet 07
Exercise sheet 08
Exercise sheet 09
Exercise sheet 10
Exercise sheet 11
Exercise sheet 12

Projects to be presented by students:
Project 1:   Sheaf cohomology and basic properties (June 16, 13.30-14.30, Jorma Dooper, Gijsbert van Vliet)
Project 2:   Introduction to spectral sequences (June 16, 14.30-15.30, Daan van der Maas, Niels van der Weide)
Project 3:   Characteristic classes (June 18, 13.30-14.30, Lauran Toussaint, Luuk Hendrikx)
Project 4:   Brown representability theorem and generalized cohomology theories (June 18, 14.30-15.30, Laura van den Bergh, Dennis Hendrikx, Merlijn Keune)

Oral exams (students should feel free to swap slots and to inform us about it):
Slot 1:   June 23, 10.00-10.40 (Daan van der Maas)
Slot 2:   June 23, 10.40-11.20 (Niels van der Weide)
Slot 3:   June 23, 11.20-12.00 (Luuk Hendrikx)
Slot 4:   June 23, 12.00-12.40 (Lauran Toussaint)
Slot 5:   June 24, 10.00-10.40 (Jorma Dooper)
Slot 6:   June 24, 10.40-11.20 (Gijsbert van Vliet)
Slot 7:   June 24, 11.20-12.00 (Laura van den Bergh)
Slot 8:   June 24, 13.00-13.40 (Dennis Hendrikx)
Slot 9:   June 24, 13.40-14.20 (Merlijn Keune)