Algebraic Topology I Autumn 2024

Lecturer
Paul Biran
Coordinator
Giovanni Ambrosioni

General informations and contents

  • Content. This is an introductory course in algebraic topology, which is the study of algebraic invariants of topological spaces. The main topics we will cover are singular homology, cell complexes and cellular homology and the Eilenberg-Steenrod axioms. We will assume a basic knowledge of point-set topology and group theory. We encourage students to familiarise with the concepts of covering space and fundamental group (as treated for instance in Hatcher's or Bredon's books in the literature).
  • Lectures. Lectures will take place on Wednesdays between 10.15 and 12.00 and Fridays between 14.15 and 16.00 in HG E 1.1. Recordings of all the lectures will be provided at the following link.
  • Exercise classes and Forum. In this course we won't have weekly exercise sessions, but some lectures during the semester will be replaced by exercise sessions led by the course organiser. Apart from this, there is the possibility of asking questions on the DMATH forum: there you will find a post dedicated to each exercise sheet, where you can ask (and answer!) questions related to exercises, but don't hesitate to create a 'new topic' if you have any other math-related question. Suggestions for topics to discuss during the exercise classes are very welcome!
  • Old notes and recordings. You can find recordings and handwritten notes of a past edition of the lecture at this link.

Weekly contents

Week Content
18/20.09 Homotopy - definitions, basic constructions and properties.
25/27.09 Singular homology - definition, basic properties and first calculations.
02/04.10 Hurewicz theorem and a quick introduction to homological algebra - chain complexes and exact sequences.
09/11.10 Homology long exact sequence, relative homology and reduced homology
16/18.10 Axiomatic approach to homology, singular homology with coefficients
23/25.10 The homology of the sphere and related calculations, degrees of maps and applications (Brouwer fixed-point theorem)
30.10/1.11 Calculation of degrees, local degrees
06/08.11 More about degrees and applications
13/15.11 CW-complexes and exercise class/td>

Exercises

Exercise sheets and their solutions will be posted here. We will roughly have one exercise sheet a month. Note that exercise sheets will not be graded, but if you have questions feel free to use the dedicated post on the forum or directly email the organiser.
Problem sets Solutions
Problem set 1 Solution 1
Problem set 2 Solution 2
Problem set 3 Solution 3
Problem set 4 Solution 4

Literature