Introduction

This course covers topics to be selected from topology, the foundation of modern geometry and analysis. This year, we will develop the theory of homology, which uses sophisticated linear algebra techniques to understand topological spaces. The course will be held in English. Core topics include homotopy equivalences, simplicial homology, and singular homology. Additional topics will be considered as time allows.

Prerequisite

I'll assume that you have a reasonable understanding of topology on an undergraduate level. You'll also want to have a strong level of comfort with linear algebra, vector spaces, and (more generally) abelian groups.

Textbook and literature

I'll work out of two books: Elements of algebraic topology by James Munkres, and Algebraic topology by Allen Hatcher. There is a lot of overlap between the two, but each book covers certain topics in a more approachable manner. Hatcher's book has the advantage that it is freely available for download from his webpage. I will also provide detailed course notes on the e-classroom.

Topics

Homology is one of the main branches of algebraic topology. The older name for algebraic topology is combinatorial topology, and we'll start the course by giving combinatorial models for topological spaces: simplicial complexes, Δ complexes, and cell complexes. We'll also discuss homotopy equivalence, a weaker notion of two topological spaces being "similar" than that of homeomorphism.

We'll then develop the definition and some intuition for homology. This will involve almost as much linear algebra (and the analogue to linear algebra over the integers) as topology.

As time allows, we'll consider additional topics, such as connections to combinatorics.

Exams and grading

The course will adjust to the COVID19 pandemic, and will be held over Zoom and/or in-person as the states of the pandemic and ourselves allow. Note that we will wear masks and have windows open for any in-person activities.

Homework assignments will comprise 20% of your grade. The remaining 80% of your grade will be determined by exams.

There will be 1 midterm examination (kolokvij), to be held at a mutually convenient time midway through the semester.

In accordance with university policy, there will be at least 4 opportunities to take an instance of the final. (However, you may only take the final once more after you have attained a passing grade.)

The exam portion of your grade will be made up either of midterm and final, or else of final alone. More specifically, your overall grade will be determined by

20%(homework) + max( 30%(midterm) + 50%(final), 80%(final) )

You may bring one hand-written A4-sized sheet of paper with you to each exam.

I strongly recommend that you study for and take the midterm exam, even if you think it is unlikely you'll do well on it.

Collaboration

I expect much of your learning to take place in working out the homework problems. While you may collaborate with other students on the homework, I expect you to have thought hard about the problems on your own first. If you do collaborate substantially, then you should so indicate on your homework paper.
Your write-up should in any case represent your own solutions, written in your own words. Copying the solutions from another student or from an internet source is a form of academic dishonesty, and will not be tolerated.

Obviously, exams will be strictly your own work.