Topology

Topologie

6 EC

Semester 2, period 4, 5

5122TOPO6Y

Owner Bachelor Wiskunde
Coordinator dr. Maximilian Engel
Part of Exchange Programme Faculty of Science, specialisation BSc Mathematics, year 1Minor Mathematical Themes, year 1Bachelor Mathematics, year 2Double bachelor Mathematics and Physics, year 2Double bachelor Mathematics and Computer Science, year 2Bachelor Bèta-gamma, major Wiskunde, year 3Bachelor Econometrics and Data Science, specialisatie Data Science, year 3Bachelor Econometrics and Data Science, specialisatie Econometrics, year 3
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Course manual 2024/2025

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

  • Algemene Topologie: de student kent definities, voorbeelden en basiseigenschappen van de volgende concepten: topologie, open, gesloten, sluiting, basis, Hausdorff (H), compact (C), samenhangend (S), padsamenhangend, continue afbeeldingen, homeomorfisme, deelruimte-topologie, product-topologie, quotiënt-topologie.
  • Algebraische Topologie: de student kent definities, voorbeelden en basiseigenschappen van de volgende concepten: path, homotopie, retractie, overdekking, liften van afbeeldingen, de fundamentaalgroep, enkelvoudig samenhangend.
  • Stellingen: Studenten kunnen de volgende belangrijke stellingen formuleren, bewijzen en toepassen: plaklemma, gedrag van H/S/C onder producten en continue afbeeldingen, samenhangende en compacte deelverzamelingen in R, karakterisatie van de quotiëntafbeelding, classificatie van oppervlakken, fundamentaalgroep van de cirkel en enkele oppervlakken. Brouwer’s dekpuntstelling, de hoofdstelling van de algebra.
  • De student kan verschillende topologische eigenschappen van topologische ruimten nagaan en bewijzen.
  • De student begrijpt het belang van topologische technieken bij het bestuderen van problemen en de invloed van topologische eigenschappen op het gedrag en het bestaan van oplossingen.

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 cannot 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, it can be decided to deny the positive effects of homework to be counted towards the final grade of the course. In case of personal circumstances, as described in OER-A Article A-6.4, a different 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

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 (section 55)

Week 13: Homotopy type and deformations (section 58).

Week 14: The fundamental group of some surfaces (sections 59 and 60).

Week 15: Review

Honours information

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

Contact information

Coordinator

  • dr. Maximilian Engel

Teachers

  • Maximilian Engel

Staff

  • Quirijn Boeren BSc
  • Giacomo Grevink BSc
  • Jelle Groot
  • Annika Holtrup
  • Gerrit Oomens