# Algebraic Topology

Algebraic Topology

## (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.

• Lecture Notes for the courses are here.
• The Problem Sheets are here. Solutions to the Problem Sheets are here. (You need a password to enter.)
• NEW: I now have a dedicated forum site for comments and questions (ETH/UZH login only.)

## (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.

• Lecture Notes for the course are here.
• The Problem Sheets are here. Solutions to the Problem Sheets are here. (You need a password to enter.)

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.