Algebraic Topology I Autumn 2022
 Lecturer

Sara Kalisnik Hintz
 Coordinator

Alessio Cela
Lectures
The lectures will take places on Wednesdays at 10:1512:00 and on Fridays 14:1516:00 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 EilenbergSteenrod axioms
Prerequisites
You should know the basics of pointset 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 16 and Section 8 in G. Bredon, "Topology and geometry", Graduate Texts in Mathematics, 139. SpringerVerlag, 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 MayerVietoris (part 1)
Lectures 18  MayerVietoris (part 2).
Lecture 19: MayerVietoris (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 17:0018:00 (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.
Old Exams
Literature