# Algebraic Topology I Autumn 2022

Lecturer
Sara Kalisnik Hintz
Coordinator
Alessio Cela

## 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. There is a forum which can be used for discussions about the course.

## 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. 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.

## Lectures Content

• Lecture 1 - Homotopy - definitions, basic constructions and properties.
• Lecture 2 - (Strong) Deformation Retractions, Operations with Homotopies.
• Lecture 3 - Homotopy Groups and Delta Complexes. A table of homotopy groups of spheres.
• Lecture 4 - Simplicial homology.
• Lecture 5: Simplicial homology and Singular homology
• Lecture 6 - Functorial properties of Simplicial homology and augmentation.
• Lectures 7 and 8 - Hurewicz Theorem and homotopy invariance of homology.
• Lectures 9 and 10 - Homological algebra.
• Lectures 11 and 12 - Relative homology and splittings.
• Lecture 13: relative homology (see notes from previous lectures 11 and 12) and Excision (part 1). See here for the Clicker Question.
• Lectures 14 - Excision (part 2).
• Lectures 15 - Excision (part 3).
• Lectures 16 - Excision (part 4).
• Lecture 17: Invariance of dimension and Mayer-Vietoris (part 1)
• Lectures 18 - Mayer-Vietoris (part 2).
• Lecture 19: Mayer-Vietoris (Part 3), equivalence between simplicial and singular homology (Part 1) Here are the Clicker Questions: n.1, n.2, n.3
• Lecture 20: equivalence between simplicial and singular homology (Part 2) and Axioms for homology (part 1)
• Lecture 21 - Degree of Maps between spheres (Part 1).
• Lecture 22 - Degree of Maps between spheres (Part 2).
• Lecture 23 - Degree of Maps between spheres (Part 3).
• Lecture 24: Degree (Part 4) and CW complexes (part 1)
• 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) , Exercise Session .
• Lecture 28: watch this video Gunnar Carlsson on the Shape of Data.
• ## Exercises

The new exercises will be posted here once every two weeks (the first one being on the 26th September and the last one on the 5th December). We will also offer regular office hours, during which you can ask questions. Office hours will take place in the weeks after a new exercise sheet is posted, on Tuesdays at 17:00-18:00 (the first one being on the 4th October and the last one on the 13th of December) in HG G 26.1. If possible, please ask questions in advance, via email to alessio.cela@math.ethz.ch. Solutions to the exercises will also appear in the weeks after the exercises are posted.

exercise sheet solutions
Exercise sheet 1 Solutions 1
Exercise sheet 2 Solutions 2, Details ex 2, 3 and 5
Exercise sheet 3 Solutions 3
Exercise sheet 4 Solutions 4
Exercise sheet 5 Solutions 5
Exercise sheet 6 Solutions 6
Exercise sheet 7 Solutions 7
Exam Preparation (updated version)

## Old Exams

Exam solutions
Exam HS15 Solutions HS15
Exam FS16 Solutions FS16
Exam HS18 Solutions HS18