Avsnittsöversikt

►Lecture 1

  • Course content:

    The course treats the foundations of general set topology (topological spaces, continuity, compactness, connectedness, quotient topologies), the fundamental group and the classification of closed surfaces.

    Teachers

    Lectures: Gregory Arone  
     E-mail: gregory dot arone at math dot su dot se

    Exercise sessions: Coline Emprin  
     E-mail: Coline dot Emprin at math dot su dot se 

    Course literature

    John M. Lee, Introduction to Topological Manifolds, 2nd edition. (Springer)

    As extra literature one may use J.R. Munkres, Topology (Prentice Hall) and A. Hatcher, Algebraic topology (freely available online).

    Schedule in TimeEdit

    We meet: Mondays 13:00 - 16:00
    Except for the first and last lecture, the time will be divided between:
    Exercises: 13:00 - 14:00.
    and
    Lectures: 14:00 - 16:00

    Note that the lecture-plan below is tentative and might change slightly.

    Howework assignments

    A homework assignment will be posted once in two weeks Monday, with deadline Thursday the following week. The homework assignment is discussed at the exercise session after the deadline. The homework gives up to 3 bonus points on the exam (and on the following re-exam).

    You are allowed to collaborate on the homework assignments, and you also are allowed to ask for help from the teachers, after making an effort to do them yourself. But you should write the assignments individually and not copy. Please do not seek help with the assignments on the internet and do not use ChatGPT or similar, not least because it often gives a wrong answer. The main purpose of the assignment is to help you master the material. The bonus points are just a bit of extra motivation. If you cheat, you will be cheating yourself.

    Written exam: December 15 (2025), 8:00 - 13:00 (place tba).
    Do not forget to register for the exam (at the latest 10 days before the exam)!

    The exam (and re-exam) will consist of 5 problems giving at most 30 points. 

    The grading for the written exam: E 16, D 19, C 22, B 25, A 28. (may be adjusted slightly)

    One of the  following five problems will appear on the exam (and re-exam)
    (I will roll a dice to determine which):

    1. Closed map lemma
    First prove Proposition 4.36a and 4.36b. Then prove Lemma 4.50.

    2. The fundamental group is a homotopy invariant
    Prove first Lemma 7.45 then Theorem 7.40.

    3. Attaching a disc
    Prove Proposition 10.13.

    4. Existence of the Universal Covering Space
    Show Step 1, 3 and 4 of Theorem 11.43.

    5. Classification of coverings
    Prove Theorem 12.18, but assume it to be known that the map
    "hat q", defined in the proof, is a covering map.

    In the proofs you are allowed to use any result of the book
    that appears earlier, as long as you write out the statement
    of the result you are using (for instance, if you want to
    use Proposition 2.15 then you write "We know from earlier
    that: A map between topological spaces is continuous if and
    only if the preimage of every closed subset is closed.").

    Note that the proofs of 3,4,5 do not need to be as detailed as in the
    book, but need to contain an explanation of all steps.

    Note also that there is an errata with a (for instance) small correction to the
    proof of 3.

    Written  re-exam: tba.
    Do not forget to register for the exam (at the latest 10 days before the exam)!

    Exam returns take place during the student affairs office's opening hours Tuesdays 11:45-12:45.
    Before you come and pick up your exam, you need to fill in this form: https://survey.su.se/Survey/51990/en. When your exam is ready for return, you will receive an email from tentataerlamning@math.su.se.

    Examination Rules at the Department of Mathematics


    Resources
►Lecture 1