# Algebraic Topology I Autumn 2023

Lecturer
Sara Kalisnik Hintz
Coordinator
Aitor Iribar Lopez

## Lectures

The lectures will take places on Wednesdays at 10:15-12:00 and on Fridays 14:15-16:00 in HG E 1.1.

## Content

This is an introductory course in Algebraic Topology. Topics covered are:
• Singular homology
• Cell complexes and cellular homology
• The Eilenberg-Steenrod axioms

## Prerequisites

You should know the basics of point-set topology. It is useful to have (though not absolutely necessary) basic knowledge of the fundamental group and covering spaces (at the level covered in the course "topology"). Students not familiar with this topic can look this up, for example in Chapter 3, Section 1-6 and Section 8 in G. Bredon, "Topology and geometry", Graduate Texts in Mathematics, 139. Springer-Verlag, 1997 ". (Members of ETH can legally download the ebook trough the ETH network.) Some knowledge of differential geometry and differential topology is useful but not strictly necessary.

## Content of the Lectures

• Lecture 1 - Homotopy - Quotient topologies and the definition of an homotopy.
• Lecture 2 - Retractions and deformation retractions. Operations with homotopies.
• Lecture 3 - Review of the fundamental group and higher homotopy groups. Delta complexes. Homotopy groups of spheres
• Lecture 4 - Delta complexes and chain complexes.
• Lecture 5 - Simplicial homology.
• Lecture 6 - Singular homology.
• Lecture 7 - Functorial properties of Simplicial homology and augmentation, 0-dimensional homology groups.
• Lecture 8 - Hurewicz Theorem, part 1.
• Lecture 9 - Hurewicz Theorem, part 2, Homotopy Invariance. See here for the clicker questions.
• Lecture 10 - Homotopy Invariance, Homological Algebra, part 1.
• Lecture 11 - Homological Algebra.
• Lecture 12 - Homological Algebra, part 2, Relative Homology, part 1.
• Lecture 13 - Relative Homology, part 2 , clicker questions
• and extra exercise .
• Lecture 14 - Split Sequences of Chain Complexes and Groups.
• Lectures 15 - Excision (part 1).
• Lectures 16 - Excision (part 2).
• Lecture 17 - Excision (part 3), Mayer Vietoris Sequence.
• Lectures 18 - Mayer-Vietoris Sequence (exercise session).
• Lecture 19 - Mayer-Vietoris Sequence (part 2), Equivalence of Simplicial and Singular Homology.
• Lecture 20 - Equivalence of Simplicial and Singular Homology. (part 2), Axioms for homology .
• Lectures 21 - Degree (part 1).
• Lectures 22 - Degree (part 2).
• Lectures 23 - Degree (part 3).
• Lecture 24 - Degree (part 4) , CW complexes.
• Lecture 25 - CW complexes and Cellular Homology (part 1).
• Appendix: Homology and path components , the relative case .
• Lecture 26 - CW complexes and Cellular Homology (part 2).
• Lecture 27 - CW complexes and Cellular Homology (part 3).
• Lecture 28 - A short introduction to Topological Data Analysis.

## Exercises

The new exercises will be posted here once every two weeks (the first one being on the 4th of October and the last one on the 13th of December). Solutions to the exercises will also appear in the weeks after the exercises are posted.

We will also offer regular office hours, during which you can ask questions and we will solve a couple exercises similar to those in the exercise sheets. Office hours will take place every Tuesday, until the 19th of December, from 16:00 to 17:00, in room CHN D.46. In addition to the problem sheets, we will upload some extra exercises, whose solutions will not be posted, but some of them will be discussed during the consultation sessions. If possible, please ask questions in advance, via email to aitor.iribarlopez@math.ethz.ch.

More exercises can be found here:
• Hatcher's Algebraic Topology book - there are exercises at the end of each section.
• Community Solutions - several old exams for algebraic topology are posted here with solutions.
• ## Old Exams

exam solution
Exam HS15 Solutions HS15
Exam FS16 Solutions FS16
Exam HS18 Solutions HS18