## Algebraic Topology II

## (Spring 2018)

This is a continuation course to Algebraic Topology I.

Topics covered include:

- universal coefficients,
- products in homology (the Eilenberg-Zilber Theorem and the Künneth Formula),
- the cohomology ring,
- fibre bundles, the Leray-Hirsch Theorem, and the Gysin sequence,
- topological manifolds and Poincaré duality,
- higher homotopy groups, fibrations and cofibrations,
- classical theorems in homotopy theory (time permitting).

You should be familiar with all the material from Algebraic Topology I, but there are no other specific prerequisites.

## Algebraic Topology I

## (Autumn 2017)

This was be an introductory course in algebraic topology, intended for upper-level undergraduates and beginning graduate students.

Topics covered included:

- the fundamental group,
- singular homology,
- cell complexes and cellular homology,
- the Eilenberg-Steenrod axioms.

Along the way we will introduced the basics of homological algebra and category theory.

This is a student seminar intended for students who are familiar with the material from Algebraic Topology I and are either taking (or have taken in a previous year) Algebraic Topology II. The course is being jointly given by me and Jagna Wiśniewska.

The course studies vector bundles in algebraic topology, concentrating on two main topics: topological K-Theory and characteristic classes.

Topics covered include:

- vector bundles over topological spaces,
- topological K-theory,
- Bott Periodicity,
- characteristic classes.

**EDIT: **Jagna now has her own webpage, so the course webpage has shifted to her site. You can find it here.