Algebraic Topology I Autumn 2023
 Lecturer
 Sara Kalisnik Hintz
 Coordinator
 Aitor Iribar Lopez
Lectures
The lectures will take places on Wednesdays at 10:1512:00 and on Fridays 14:1516:00 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 EilenbergSteenrod axioms
Prerequisites
You should know the basics of pointset 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 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.
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, 0dimensional 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  MayerVietoris Sequence (exercise session).
Lecture 19  MayerVietoris 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
Literature