Will J. Merry

Algebraic Topology

Algebraic Topology

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.

  • Lecture Notes for the course are here.

  • The Problem Sheets are here.

  • Solutions to the Problem Sheets are here. (Edit: Not anymore!)

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.

  • Lecture Notes for the courses are here.

  • The Problem Sheets are here.

  • Solutions to the Problem Sheets are here. (Edit: Not anymore!)

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.

Vector Bundles in Algebraic Topology (Spring 2018)