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

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.

Syllabus:

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.