Martin David Palmer -- Lecture course -- Algebraic Topology I
Algebraic Topology I
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V4D1 — Algebraic Topology I

Lecturer: Dr Martin Palmer-Anghel
Assistent: Dr Benjamin Böhme
 
Wintersemester 2018/2019
Lectures: Monday 14–16, Wednesday 8–10
(First lecture: Monday 8th October 2018)
Room: Zeichensaal (Wegelerstr. 10)
Link to the official course webpage in basis.

Topics

This is a continuation of the course Topology II in Sommersemester 2018, focused on homotopy theory. The main topics will be the following (*subject to modification as the lectures progress!*).

  • A brief recollection of some homotopy theory that was covered in the lectures Topology I and Topology II.
  • Fibrations and cofibrations, homotopy (co)fibres.
  • The Blakers-Massey theorem and the Freudenthal suspension theorem.
  • Brown representability and Eilenberg-MacLane spaces.
  • Postnikov and Whitehead towers, k-invariants.
  • Quasifibrations and the Dold-Thom theorem.
  • Serre classes and rational homotopy groups of spheres.
  • Principal bundles, vector bundles, classifying spaces.
  • A brief introduction to characteristic classes (see also this parallel seminar on characteristic classes). Bordism and the Pontrjagin-Thom theorem.

Here is a cumulative outline of the topics covered in the lectures so far:

Literature

  • Tammo tom Dieck, Algebraic Topology
  • Allen Hatcher, Algebraic Topology
  • John Milnor and Jim Stasheff, Characteristic Classes
  • Robert Switzer, Algebraic Topology — Homotopy and Homology
  • George W. Whitehead, Elements of Homotopy Theory

Exercises

There are two exercise classes, taking place once per week, starting in the second week of the semester.

  • Friday 10–12 — Daniel Brügmann — seminar room 1.008.
  • Friday 12–14 — Benjamin Ruppik — seminar room 0.011.

The exercise sheets will be uploaded here on Fridays. The general pattern is that exercise sheet n (uploaded on the Friday of the nth week of the semester, for positive n) should be handed in before the lecture on Monday afternoon of week n+2. For the precise schedule, see below. Exercise sheets may be handed in jointly by at most three students.

  • Exercise sheet 0 — Not to be handed in. — Discussed in classes in week 2.
  • Exercise sheet 1 — Due on Monday, 22 October 2018. — Discussed in classes in week 3.
  • Exercise sheet 2 — Due on Monday, 29 October 2018. — Discussed in classes in week 4.
  • Exercise sheet 3 — Due on Monday, 5 November 2018. — Discussed in classes in week 5.
    • Erratum — correction to exercise 4a of sheet 3.
  • Exercise sheet 4 — Due on Monday, 12 November 2018. — Discussed in classes in week 6.
  • Exercise sheet 5 — Due on Monday, 19 November 2018. — Discussed in classes in week 7.
  • Exercise sheet 6 — Due on Monday, 26 November 2018. — Discussed in classes in week 8.
  • Exercise sheet 7 — Due on Monday, 3 December 2018. — Discussed in classes in week 9.
    • Corrected version (28.11.2018) — statement of exercise 3 clarified, misleading hint and remark removed from exercise 4b, simplifying assumption added for exercise 4.
  • Exercise sheet 8 — Due on Monday, 10 December 2018. — Discussed in classes in week 10.
    • Corrected version (04.12.2018) — definition of Moore loop space corrected in exercise 1.
  • Exercise sheet 9 — Due on Monday, 7 January 2019. — Discussed in classes in week 12.
  • Exercise sheet 10 (uploaded 4 January) — Due on Monday, 14 January 2019.
  • Exercise sheet 11 (uploaded 11 January) — Due on Monday, 21 January 2019.
  • Exercise sheet 12 (uploaded 18 January) — Due on Monday, 28 January 2019.
  • Exercise sheet 13 (uploaded 25 January) — Not to be handed in (revision sheet).
  • Optional extra exercises (updated 26.11.2018) — Not to be handed in.

Exams

The exams will be written. The dates are the following.

  • First exam: 9:00–11:00, Wednesday 6 February 2019, Kleiner Hörsaal, Wegelerstr. 10.
  • Klausureinsicht (exam review): 14:00–15:00, Thursday 7 February 2019, seminar room 0.011.
  • Second exam: 9:00–11:00, Wednesday 13 March 2019, Kleiner Hörsaal, Wegelerstr. 10.
  • Klausureinsicht (exam review): 14:00–15:00, Thursday 14 March 2019, seminar room 0.011.

As usual, the requirements for admission to the exam are:

  • Obtaining at least half of the credits from the exercise sheets (from sheets 1–11, each of which has 20 points available, so the threshold is at least 110/220 points).
  • Presenting at least two exercises on the blackboard during the exercise sessions.