This course is optional for mathematics students. It will probably be the first contact of the student with the methods of Algebraic Topology. These ideas are incredibly important in Differential Geometry, Algebraic Geometry, and Topology, and they find also application in other areas like Analysis, Algebra, or Logic. The course is optional, but recommended to any student interested in pure Mathematics. Please find more information about the study advisory paths in the bachelor at the student website.
Leerdoelen:
Following the steps of Inleiding Topologie, our goal in this course will be to continue studying topological spaces. To do so, we will need new tools:
- We will define homotopies, homotopy equivalences, and the fundamental group.
- We will compute the fundamental group of the circle.
- We will introduce the van Kampen theorem, which allows us to compute the fundamental group for more complicated topological spaces.
- We will relate the fundamental group to the theory of free group actions and covering spaces.
- If time allows, we will define homology. This is an algebraic gadget that is suitable for detecting higher dimensional properties of our topological spaces. We will put emphasis on its applications.
A key part of the course is for the student to become familiar with more examples of topological spaces. These will serve as a testing ground for the techniques we introduce. Part of the course will be dedicated to studying and classifying surfaces, both in the oriented and non-oriented cases.
At the end of this course, a student is able to:
- Prove elementary results about homotopies between maps and homotopy equivalences between topological spaces, and illustrate these notions through examples.
- Compute the fundamental group for a large class of spaces.
- State the classification of surfaces, as well as the key ideas involved in its proof.
- Explain the terms "connected sum", "orientable" and "Euler characteristic" and check the later two in examples.
- Work with covering spaces, prove the standard results about them, and compute them in simple cases.
- Define the notion of homology, and compute it for some small familiar examples.
- Decide for many pairs of spaces whether they are homotopy equivalent (or even homeomorphic) or not.
- Provide applications inside and outside of Topology (Brouwer fixed point theorem, Borsuk-Ulam theorem, fundamental theorem of algebra).
Exercise sheets, lecture notes, and general announcements regarding the course will appear on Blackboard.
Onderwijsvormen:
Two times per week two hours of lectures and two times per week tutorials.
Toetsing:
There are two exams for this class, the midterm will count for 30% and the final exam will count for 50% of the final grade.Furthermore, there is homework which counts for 20% of the final grade.
In case of a retake exam, the retake exam counts for 80% and the homework for 20% of the final grade, while the original grades of midterm and final exam are deleted.
Herkansing en inspanningsverplichting:
Students with a final grade lower than 4 are eligible to do the retake exam only if they have handed in solutions to all hand-in homework problems, either before or after the final exam.
Taal van het vak:
The language of instruction is English.
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