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# Topology and geometry

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We will start with a quick  review on the use of topology to solve results that make no reference to topology in their statements. Then we will delve into the basics of Algebraic Topology: homotopy, homotopy equivalence, CW complexes and fundamental group. The fundamental group of a topological space will occupy a major part of this course: firstly we will develop computational tools, such as van Kampen theorem and then study the relationship between fundamental group and covering spaces. The path lifting property for covers will be used to compute the fundamental group of S^1, and we will use van Kampen's theorem to get presentations of the fundamental groups of any CW-complex. On our way, we will also prove a correspondence between covers and subgroups of the fundamental group.

We will use the fundamental group to state and prove the classification of compact surfaces.

The second and smaller part of the course will deal with homology. Homology is a "higher dimensional" and "Abelianized" version of the fundamental group. We will study its behaviour with respect to homotopies and introduce its fundamental properties and computational tools such as Mayer—Vietoris, excision and relative homology.

Prerequisites:
A good understanding of the notion of topological space and of continuous maps.
Some familiarity with the basic operations on topological spaces:
products, quotients, disjoint unions.

Exams forms:
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.
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## Learning outcomes

At the end of this course, a student is able to:
• Define CW-complexes. Give some examples and counterexamples.
• Define homotopy between maps, homotopy equivalence between topological spaces and illustrate these notions by some examples.
• Define the fundamental group and compute it for a large class of spaces.
• State van Kampen's theorem and sketch its proof.
• State the classification of surfaces. Explain the terms "connected sum", "orient able" and "Euler characteristic".
• Define (universal) covers, state the main classification theorems about covers, and the universal property of universal covers.
• State and prove the path lifting property for covers.
• Define the notion of homology, and compute it for some small familiar examples.

## Prior knowledge

WISB124 Introduction to groups and rings and WISB243 Introduction to topology. See the courseplanner (cursusplanner.uu.nl) for the contents of those courses: select Faculty of Science and then the programme of the bachelor Mathematics of the most recent year.

If anything remains unclear, please check the FAQ of Utrecht University.

## Offering(s)

• ### Start date

5 februari 2024

• Ends
19 april 2024
• Term *
Period 3
• Location
Utrecht
• Instruction language
English
• Register between
30 Oct, 23:59 - 24 Nov 2023
Enrolment starts in 149 days
These offerings are valid for students of TU Eindhoven