Algebraic Topology

Algebraic Topology


Algebraic Topology I (Autumn 2017)

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

Topics covered include:

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

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

  • Lecture Notes for the course are here.
  • The Problem Sheets are here.
  • Here is a page intended for current students enrolled in the class. It contains the solutions to the Problem Sheets and other miscellaneous information. (You need a password to enter.)

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),
  • cohomology (the ring structure and Poincaré duality),
  • higher homotopy groups, fibrations and cofibrations,
  • classical theorems in homotopy theory (the Hurewicz Theorem, the Blakers-Massey Theorem and and the Whitehead Theorem.)

I will again produce full lecture notes.


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—the Euler class, Stiefel-Whitney classes, Chern classes, Pontryagin classes,
  • characteristic classes as obstructions.

Vector Bundles in Algebraic Topology (Spring 2018)