Course manual 2021/2022

Course content

In this course the fundamentals of topology are treated. Various concepts that play a role in the analysis are reflected in this course in abstract form. In this course we lay a foundation for the further study of geometry, algebraic topology and differential topology. Topics covered are topological spaces, continuous maps and homeomorphisms, connectedness, compactness, and quotient spaces. The fundamental group will be discussed in detail. We calculate the fundamental group of a number of known spaces and study the relationship between fundamental groups and covering spaces.

Study materials

Literature

J.R. Munkres, 'Topology', Prentice Hall Inc., 2nd edition, 2000, ISBN 0-13-181629-2.

Objectives

The knowledge domains are: 1. Topological spaces and continuous maps, 2. Compactness and connectedness, 3. Equivalence relations and quotient spaces, 4. Paths, homotopies, and retractions, 5. Covering spaces and lifts, 6. The fundamental group. 1. In each domain area, the student can formulate and apply the definitions.
In each domain area the student can determine if a space or map has the relevant properties
In each domain area the student can explain the interdependencies of the ideas
In each domain area the student can generate examples and counterexamples illustrating the key concepts, both mathematically and with appropriate figures
Students can determine distinguishing features of different topological spaces
6. Students can describe in writing the flow and interdependencies of ideas across chapters

Teaching methods

Self-study
Lecture

Additional materials provided on Canvas

Learning activities

Activiteit	Aantal uur
Hoorcollege	30
Tentamen	3
Tussentoets	3
Werkcollege	28
Zelfstudie	104

Attendance

Programme's requirements concerning attendance (OER-B):

Each student is expected to actively participate in the course for which he/she is registered.
If a student can not be present due to personal circumstances with a compulsory part of the programme, he / she must report this as quickly as possible in writing to the relevant lecturer and study advisor.
It is not allowed to miss obligatory parts of the programme's component if there is no case of circumstances beyond one's control.
In case of participating qualitatively or quantitatively insufficiently, the examiner can expel a student from further participation in the programme's component or a part of that component. Conditions for sufficient participation are stated in advance in the course manual and on Canvas.
In the first and second year, a student should be present in at least 80% of the seminars and tutor groups. Moreover, participation to midterm tests and obligatory homework is required. If the student does not comply with these obligations, the student is expelled from the resit of this course. In case of personal circumstances, as described in OER-A Article A-6.4, an other arrangement will be proposed in consultation with the study advisor.

Additional requirements for this course:

Aanwezigheid bij de werkcolleges is verplicht. Als je niet bij minstens 80% van de werkcolleges aanwezig bent geweest dan vervalt je recht op het hertentamen, zoals vermeldt in het OER-B artikel 4.9 lid 2.

Assessment

Item and weight	Details
Final grade
1 (100%) Deeltoets

The midterm test covers point-set topology and counts for 25%

The Final Exam covers both point-set topology and algebraic topology and counts for 60%

The homework counts for 15%

Fraud and plagiarism

The 'Regulations governing fraud and plagiarism for UvA students' applies to this course. This will be monitored carefully. Upon suspicion of fraud or plagiarism the Examinations Board of the programme will be informed. For the 'Regulations governing fraud and plagiarism for UvA students' see: www.student.uva.nl

Course structure

Here is a tentative overview of the course with the corresponding sections from the text by Munkres that will be covered.

Week 1: Topological spaces, examples of topological spaces, basis for a topology, relationship with metric spaces (sections 12, 13, 20).

Week 2: Product topology, subspace topology, interior and closure (sections 15,16,17).

Week 3: Hausdorff spaces, continuous functions (sections 17 and 18).

Week 4: Connected spaces (sections 23, 24, 25)

Week 5: Compactness (sections 26, 27).

Week 6 Quotient topology (section 22).

Week 7: Review and (possibly) application on surfaces: cut and paste (section 80)

Week 8: Midterm test

Week 9: Homotopy of continuous images and of paths (section 51).

Week 10: The Fundamental Group and Cover Spaces (sections 52 and 53).

Week 11: The fundamental group of the circle (section 54).

Week 12: Retractions (sections 55.1-6, 58.1-3)

Week 13: The fundamental group of some surfaces (section 59.60)

Week 14: The universal covering of a surface and the fundamental group

Week 15: Review

Timetable

The schedule for this course is published on DataNose.

Honours information

There is an honors extension worth 3 EC for the Topology course.
In the Honors extension, students will apply surface classification themselves.

Processed course evaluations

Below you will find the adjustments in the course design in response to the course evaluations.

Owner	Bachelor Wiskunde
Coordinator	prof. dr. Jo Ellis-Monaghan
Part of	Bachelor Wiskunde, year 2 Dubbele bachelor Wis- en Natuurkunde, year 2 Dubbele bachelor Wiskunde en Informatica, year 2 Bachelor Bèta-gamma, major Wiskunde, year 3

Topology