Topology and geometry

This course is optional for mathematics students. The course is recommended to students interested in pure mathematics, particularly those interested in Algebraic Topology, Algebraic and Differential Geometry, and Logic. It is also recommended to students interested in Mathematical Physics (for instance, in General Relativity or String Theory).

Leerdoelen:
The course is a follow-up of Inleiding Topologie and serves two purposes:

• To introduce the student to the Homotopy Theory/Algebraic Topology approach to the study of topological spaces.
• To introduce the student to the formalism of Category Theory, which is the organising principle behind a large part of modern mathematics.

On the side of Topology, we will define homotopies, homotopy equivalences and fundamental groups. The main idea is that we are interested in computing the “holes” that a topological space may have and, indeed, many invariants in topology amount to counting holes in some suitable way. Two homotopy equivalent spaces have the same holes.

The fundamental group will occupy a major part of the course. It measures how curves in a space “wrap around holes”. Two key techniques to compute the fundamental group will be introduced: the Theorem of van Kampen and the theory of covering spaces.

On the side of Category Theory, we will introduce categories, functors, and various universal properties and constructions. All relevant categorical concepts will be introduced with topological spaces in mind, so definitions will always be motivated from concrete examples (not just from Topology but also Algebra).

Lastly, we will classify two-dimensional manifolds (i.e. surfaces). These are important examples in Topology and will serve as a testing ground for the concepts we will encounter. We will also introduce cell attachment as a fundamental method to produce topological spaces.

At the end of this course, the student is able to:

• Define homotopies between maps, homotopy equivalences between topological spaces, and illustrate these notions via examples.
• Define the fundamental group(oid) and compute it for a large class of spaces.
• State the classification of surfaces.
• Explain the terms "connected sum", "orientable", and "Euler characteristic".
• Define (universal) covers and know the standard theorems and examples thereof.
• Decide for many pairs of spaces whether they are homotopy equivalent (or even homeomorphic) or not.
• Use the language of Category Theory and know how it applies to various categories of interest (topological spaces, groups, vector spaces, sets, fundamental groupoids).

Onderwijsvormen:
Two times per week two hours of lectures and two times per week tutorials.

Toetsing:
There are four homeworks, each of which accounts for 5% of the final grade, assuming the grade is higher than the grade of the final exam. The final exam accounts for the remaining 80%.
The same computation applies to the retake exam.

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 homeworks.