## Algebraic Topology I

## (Autumn 2017)

In Autumn 2017 I taught 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 introduced the basics of homological algebra and category theory.

## Algebraic Topology II

## (Spring 2018)

In Spring 2018 I taught a continuation course to Algebraic Topology I.

Topics covered included:

universal coefficients,

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 and fibrations.

The course assumed familiarity with all the material from Algebraic Topology I, but there were no other specific prerequisites.

This was a student seminar in Spring 2018 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 included:

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.