The BCAM-Severo Ochoa Course A brief introduction to classical homotopy and homology theory (more information below) will have a Pre-course on \"Introduction to basic homology and homotopy theory "Sprint course in algebraic topology" 

Speakers: Eki Gartzia, Alba Larraya (BCAM)

The contents of this course could be adjusted depending on the previous knowledge of the participants, it will be necessary to complete a Google Form about the previous knowledge for the pre-course.

The course will consist of one (online) lecture per week, starting probably at the end of January and finishing in March. The schedule will be on Wednesdays, from 12:00 to 14:00 starting on January 31st, to March 26th. 

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

  1. CW-complexes

  2. Singular Homology (basic properties, cellular homology and isomorphism of both)

  3. Singular Cohomology (basic properties, universal coefficients theorems)

  4. Products (Cup, Cap, Cross, Künneth formula)

  5. Poincaré duality

  6. Loop space, suspension, fibrations and cofibrations

  7. Homotopy groups (definitions, basic properties, long exact sequence of a fibration)

  8. Whitehead and Hurewicz theorems

Lastly, if time allows we’ll cover:

  1. de Rham cohomology. (following the first five sections of chapter 1 of [3])

  2. De Rham’s theorem


  1. Allen Hatcher, Algebraic Topology, Cambridge University Press, 2002.

  2. Phillip Griffiths, John W. Morgan. Rational homotopy theory and differential forms. Vol. 16. Boston: Birkhäuser, 1981.

  3. Raoul Bott, and Loring W. Tu. Differential forms in algebraic topology. Vol. 82. New York: Springer, 1982.



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

Speaker: Aniceto Murillo, University of Malaga

Dates: 8-12 abril, 2024.

Venue: Universidad de Zaragoza

Modern homotopy theory, located withing the broader context of model categories or, more generally, infinity categories, has recently made significant strides in enhancing our understanding of the homotopy type of topological spaces.

However, classical homotopy theory tools remain effective, and even fundamental, in various areas where homotopical methods are regularly used.

Thus, this course aims to be an introduction to the classical results of homotopy and homology theory that could be useful for graduate students in diverse mathematical fields, both in their daily work and in their future endeavors.


Homotopy Theory:

  1. Generalities (Basic notions, CW-complexes, loop and suspensión, fibrations and cofibrations,…)

  2. Homotopy groups (First properties and results, some computations, homotopy sequence of a fibration, classical results: Blakers-Massey, Hilton-Milnor and Freudentahl theorems,…).

  3. Introduction to stable homotopy theory

Homology Theory:

  1. Generalities (Steenrod axioms, singular and celular homology, first computations and aopplications, cohomology).

  2. Basic results (Künneth formula, Universal coefficentes theorems for honmology and cohomology, Hurewicz theorem,…)

  3. Duality (Poincaré duality, Spanier-Whitehead duality)

  4. Introduction to generalized homology theories.

Some bibliography:

  • A. Hatcher, Algebraic Topology, 2002, Cambridge University Press.

  • D. Barnes, C. Roitzheim, Foundations of Stable Homotopy Theory.

  • G.W. Whitehead, Elements of Homotopy Theory, GTM 61, 1978, Springer.


Registration start:
Dec 01 2023.
Registration deadline:
Mar 25 2024.