Topology

Topologie

6 EC

Semester 2, period 4, 5

5122TOPO6Y

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

Course manual 2020/2021

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 student is able to formulate and apply the definitions of topological space and continuous maps
  • The student is able to formulate and apply the definitions of compactness and connectedness and check whether a given topological space has these properties
  • The student is able to work out the quotient construction for an equivalence relation on a topological space
  • The student is able to explain the idea of homotopy and give the definition of the fundamental group
  • The student is able to explain the connection between cover maps and the fundamental group
  • The student is able to give the definition of a covering and explain the connection with the fundamental group

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.

Contact information

Coordinator

  • prof. dr. Jo Ellis-Monaghan

Docenten

  • Jo Ellis-Monaghan