Topologie I, Sommersemester 2023

Inhaltsbeschreibung / course description and syllabus

(für Einschreibeschlüssel siehe Inhaltsbeschreibung oben / see course description above for the enrolment key)

Vorlesungsskript / lecture notes
The notes for this semester's course are complete as of 21.07.2023; they include the "extra" Lecture 26 (which is non-examinable material), but nothing from the final lecture, since that was purely review.
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Ankündigungen / Announcements

  • 2.08.2023: The results of the exam on 1.08.2023 are now available. I will be available in my office (1.301 in the math building) on Thursday, 3.08 from 15:00 to 1600 for anyone who would like to look through their marked exams (i.e. Klausureinsicht).
  • 27.04.2023: If you have not yet joined the moodle for this course, please do so. If you want to attend a problem session (Übung), you must first use the moodle to sign up for Group 1 or Group 2. You similarly must use the moodle if you want to submit homework electronically, or receive occasional time-sensitive announcements about the course via e-mail.
previous announcements (no longer relevant)

Exam info (NEW)

Here is a practice exam. It has the same instructions and format that the real exam will have, though it's possible that the real exam may be slightly shorter. I passed out a version of this exam in the final lecture and discussed the problems, though I have made one minor change to it since then (Problem 5(b) is now stated slightly differently, though the solution is still almost the same as what we discussed in lecture).

As indicated in the instructions on the practice exam, the actual exam will be open book, so you will be allowed to use any non-electronic resources you bring with you (i.e. books and notes, printouts of the lecture notes, but not smartphones or laptops). Any material that was covered in the lectures in this course should be considered examinable; all of it is in the lecture notes, though the notes also contain some non-examinable material. To be precise, you may consider each of the following specific topics in the lecture notes to be off limits for the exam:

  • Everything in Lecture 1 other than the basic definitions that appear there (the rest is just for motivational purposes)
  • The notion of "total boundedness", briefly mentioned at the end of Lecture 5 as an explanation for why compactness and sequential compactness are equivalent in arbitrary metric spaces (that is material for an analysis course, not this one)
  • The (sketched) proof of Lemma 14.7, on why the connected sum of two surfaces does not depend on choices
  • Theorem 14.20 (every finitely presented group is the fundamental group of something), since we didn't have time to discuss it in any detail in lecture
  • Everything that involves smooth structures on manifolds, mainly in Lectures 18 and 19
  • Everything about bordism theory in Lecture 21; this is nice to know about for motivation, but we can't yet do anything serious with it in this course
  • The "extended digression" at the end of Lecture 23 on fundamental classes and how to visualize relative cycles; this is yet another area in which knowing something about it may be useful for your intuition, but you are not officially required at this point to know anything about fundamental classes of manifolds.
  • The last few pages of Lecture 24, after the proof of Theorem 24.9 (i.e. the stronger "chain homotopy equivalence" version of the excision theorem)
  • All of Lecture 26

Übungsblätter / Problem sets

Somebody miscalculated the fundamental group. (Hermannplatz, 8.06.2017)

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