Pre-Course | BCAM-Severo Ochoa Course | A brief introduction to classical homotopy and homology theory

Date: Wed, Jan 31 - Wed, Mar 20 2024

Location: Online

Speakers: Eki Gartzia and Alba Larraya (BCAM)

Register: Course Website

Introduction to basic homology and homotopy theory "Sprint course in algebraic topology"

This course aims to quickly introduce the basics of homology, cohomology, and homotopy groups, mainly as preparation for the advanced course that will take place in April in Zaragoza: “A brief introduction to classical homotopy and homology theory”. 

The contents of this course could be adjusted depending on the previous knowledge of the participants:

The course will consist of one (online) lecture per week, starting probably at the end of January and finishing in March. The schedule is to be decided.

We’ll more or less follow the structure of chapters 2, 3, and 4 of [2] complemented with [1]. To cover the following:

  • CW-complexes
  • Singular Homology (basic properties, cellular homology, and isomorphism of both)
  • Singular Cohomology (basic properties, universal coefficients theorems)
  • Products (Cup, Cap, Cross, Künneth formula)
  • Poincaré duality
  • Loop space, suspension, fibrations and cofibrations
  • Homotopy groups (definitions, basic properties, long exact sequence of a fibration)
  • Whitehead and Hurewicz theorems

Lastly, if time allows we’ll cover:

  • de Rham cohomology. (following the first five sections of chapter 1 of [3])
  • De Rham’s theorem


  • Allen Hatcher, Algebraic Topology, Cambridge University Press, 2002.
  • Phillip Griffiths, John W. Morgan. Rational homotopy theory and differential forms. Vol. 16. Boston: Birkhäuser, 1981.
  • Raoul Bott, and Loring W. Tu. Differential forms in algebraic topology. Vol. 82. New York: Springer, 1982.