Topology IV.b - homotopy theory

Summary

We propose an introduction to homotopy theory for topological spaces. We define higher homotopy groups and relate them to homology groups. We introduce (co)fibration sequences, loop spaces, and suspensions. We study long exact sequences. We construct Eilenberg-Mac Lane spaces.

Content

Keywords

Homotopy groups, Cofibrations and fibrations, Loop spaces and suspension, Long exact sequence, Eilenberg-Mac Lane space, Hurewicz homomorphism

Learning Prerequisites

Required courses

Topology I, II, and II, Group Theory, Rings and Fields

Recommended courses

Rings and modules

Important concepts to start the course

Fundamental group, Homology groups, cell complexes, excision in homology

Learning Outcomes

By the end of the course, the student must be able to:

• Perform elementary computations of homotopy groups
• Compare homotopy with homology groups
• Define the notions introduced in the course
• State the main theorems and prove them
• Manipulate fibrations and cofibrations
• Apply the tools developed in the course to examples

Teaching methods

ex-cathedra teaching, exercise classes

Expected student activities

Attend the lectures and exercise sessions, solve exercises, hand in homework, prepare a presentation

Assessment methods

The final grade will be assigned based on:

20% - homework in groups

20% - oral presentation during an exercise session

60% - written exam

Dans le cas de l'art. 3 al. 5 du Règlement de section, l'€™eantdécide de la forme de l'€™examen qu'€™il communique aux étudiants concernés.

Supervision

Resources

Bibliography

Algebraic Topology, Allen Hatcher

Algebraic Topology, Tammo Tom Dieck

Algebraic Topology, Edwin Spanier

Introduction to Homotopy Theory, Paul Selick

Modern Classical Homotopy Theory, Jeffrey Strom

Ressources en bibliothèque

• Algebraic Topology / Dieck
• Introduction to Homotopy Theory / Selick
• Algebraic Topology / Hatcher
• Algebraic Topology / Spanier

Moodle Link

• https://moodle.epfl.ch/enrol/index.php?id=16696