Lecture Topology I

4 hours lecture + 2 hours exercise

Marc Kegel

Summer semester 2025


Lecture: Tuesdays 13:15–14:45 in Room 1.115 (RUD 25) and Thursdays 15:15–16:45 in Room 1.013 (RUD 25)
Course number: 3314412
First lecture: 15.04.2025

Exercise: Thursdays 11:15–12:45 in Room 1304 (RUD 26)
Tutor: Theo Müller
Course number: 33144121
First exercise session: 17.04.2025

Office hours: after the lectures

Grading of exercises: For the first weeks we will not have anyone to grade the exercises. If that changes I will announce that here.

Language: The lecture is conducted in English. Of course questions, solutions to exercises and exams can also be done in German.


Announcements:

Content: A topological space is a generalization of a metric space, allowing us to discuss continuous maps in the greatest possible generality. We will first briefly deal with pointset topology, introducing methods for constructing topological spaces and investigating properties that are preserved under homeomorphisms (bijective, continuous maps with continuous inverse).

The rest of the lecture will provide an elementary introduction to the methods and results of algebraic topology. The basic idea is to distinguish topological spaces by assigning algebraic invariants (numbers, groups, etc.) that are easier to tell apart. We will often focus on the slightly simpler but still very general case where the spaces carry the combinatorial structure of a simplicial complex.

A main goal of the course is to develop the fundamental group and simplicial homology theory. Using these invariants, we will for example give a complete proof of the classification theorem for surfaces. From the beginning, we will also emphasize applications from other areas, such as geometric topology (Heegaard splittings of 3-manifolds), differential topology (hairy ball theorem), algebra (fundamental theorem of algebra, every subgroup of a free group is free), analysis (Brouwer’s fixed point theorem), gastronomy (ham sandwich theorem), and meteorology: On Earth, there always exist two antipodal points with the same temperature and humidity.

Prerequisites: First-year courses (Analysis I, II and Linear Algebra I, II) are expected, especially basic topological notions (open sets, continuity, compactness) and elementary algebra (groups, homomorphisms).

Exam: Both exams will be written. The exact dates will be discussed in the lecture and announced here.
Second exam:
Criterion for admission to the final exam: I recommend taking the exam if at least 50% of the exercises were meaningfully worked on.
How to meaningfully work on an exercise sheet is, for example, very well explained by Prof. Manfred Lehn on his homepage. In particular, plagiarizing someone else’s solution is not a meaningful way of working on an exercise.


Exercise Sheets:
Exercise Sheet 0 (pdf)
Exercise Sheet 1 (pdf)
Exercise Sheet 2 (pdf)


Table of Contents (preliminary):

1. Overview

2. Connectedness:
2.1. Definition of connectedness
2.2. Applications

3. Construction of topological spaces:
3.1. The quotient topology
3.2. Compactness and the Hausdorff property
3.3. Collapsing a subspace
3.4. Gluing spaces together
3.5. Topological groups and homogeneous spaces
3.6. Orbit spaces

4. Homotopy and fundamental group:
4.1. Homotopic maps
4.2. Construction of the fundamental group
4.3. Coverings and fundamental groups
4.4. Homotopy type
4.5. Finitely presented groups
4.6. Seifert–van Kampen theorem
4.7. Applications

5. Manifolds:
5.1. Definitions and first properties
5.2. The connected sum
5.3. Handle decompositions
5.4. Classification of surfaces
5.5. Heegaard decompositions

6. Simplicial complexes:
6.1. Triangulations
6.2. Barycentric subdivision
6.3. Euler characteristic and the Hauptvermutung
6.4. Simplicial approximation
6.5. Brouwer’s fixed point theorem

7. Simplicial homology:
7.1. Definition of homology groups
7.2. First computations of homology groups
7.3. Chain maps
7.4. Topological invariance of homology
7.5. Mayer–Vietoris sequence

8. Degree, Lefschetz number, and Euler characteristic:
8.1. Degree of a map
8.2. Homology with coefficients
8.3. Euler–Poincaré formula
8.4. Borsuk–Ulam theorem
8.5. Lefschetz fixed point theorem

Following this lecture, "Topology II" will be offered in the winter semester 2025/2026, covering CW-complexes, higher homotopy groups, more general homology theories, and cohomology theory.

Lecture Notes:

There are lecture notes in German by Ms. Carolin Wengler from my Topology I course in 2019. This year’s course will be similar, though certainly not identical. (If you find any mistakes, even minor typos, please let me know so I can correct them.) If anyone or a group of people is interested in translating these notes to English or create a new version in English, let me know. In that case, I could set up an overleaf with the old tex file for collaborative working.

Ms. Wengler’s Lecture Notes (pdf)


Literature:

This course is based on the lecture by H. Geiges, which closely follows the book by Armstrong. For pointset topology (Chapters 1–3 and parts of 4), I also recommend the book by K. Jänich. In Chapter 5, I will partly follow a paper by H. Geiges. A very popular textbook for (algebraic) topology is the book by A. Hatcher. I also highly recommend the lecture notes by S. Friedl and C. Wendl.

M. Armstrong: Basic Topology, Springer, 1983.
S. Friedl: Notes on Algebraic Topology I–IV, available online on his homepage.
H. Geiges: Topology, lecture held in WS 2009/10 at the University of Cologne.
H. Geiges: How to depict 5-dimensional manifolds, DMV report, 2017.
A. Hatcher: Algebraic Topology, available online on his homepage.
K. Jänich: Topology, Springer, 1996.
C. Wendl: Notes on Topology I, available online on his homepage.

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