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 was a continuation of the course Topology II in Sommersemester 2018, focused on homotopy theory. The main topics were the following.

• 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.

Here is a detailed outline of the topics covered in the lectures:

• Outline of lectures — updated 5 February 2019 (lectures 1–16 and 19–28).
• Lectures 17 and 18 — given by Benjamin Böhme while I was away.

Literature

• Tammo tom Dieck, Algebraic Topology
• Albrecht Dold, Halbexakte Homotopiefunktoren
• Albrecht Dold and René Thom, Quasifaserungen und Unendliche Symmetrische Produkte
• Allen Hatcher, Algebraic Topology
• J. Peter May, A concise course in algebraic topology
• John Milnor and Jim Stasheff, Characteristic Classes
• Norman Steenrod, The topology of fibre bundles
• Robert Switzer, Algebraic Topology — Homotopy and Homology
• George W. Whitehead, Elements of Homotopy Theory

Exercises

There were 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 are 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. Exercise sheets may be handed in jointly by at most three students.

• Exercise sheet 0
• Exercise sheet 1
• Exercise sheet 2
• Exercise sheet 3
• Erratum — correction to exercise 4a of sheet 3.
• Exercise sheet 4
• Exercise sheet 5
• Exercise sheet 6
• Updated version (20.11.2018) — added hints for exercise 1.
• Exercise sheet 7
• 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
• Corrected version (04.12.2018) — definition of Moore loop space corrected in exercise 1.
• Exercise sheet 9
• Exercise sheet 10
• Exercise sheet 11
• Exercise sheet 12
• Exercise sheet 13 (revision sheet)
• Optional extra exercises (updated 23.01.2019)
• All exercise sheets (concatenated): 0–12 + revision + optional.

Exams

The exams are written. The dates are the following.

• First exam: 9:00–11:00, Wednesday 6 February 2019, Kleiner Hörsaal, Wegelerstr. 10.
• Here is a list of the grade boundaries for the first exam.
• 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.
• Here is a list of the grade boundaries for the second exam.
• 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.