# MA3408 Algebraic Topology 2 - Spring 2023

Schedule Room
Lectures: Monday 12:15-14:00 KJL21
Friday 10:15-12:00 R4 L11
Exam: Oral exams 11, 15, and 16 May Schedule
Lecturer
Clover May
Office: 1202 Sentralbygg 2
Email: clover [dot] may [at] ntnu [dot] no

## What this course is about

The course introduces the basic homotopy theory of spaces (fibrations and cofibrations, homotopy groups) and covers further classical topics in algebraic topology, such as: spectral sequences (in particular the Serre spectral sequence), vector bundles and characteristic classes, and cohomology operations.

See the study handbook and the tentative plan below for more information.

## Prerequisites

We'll build on the course MA3403 Algebraic Topology 1: Homology and Cohomology. This means that I'll assume that you know about topological spaces and that you are familiar with homology and cohomology. I'll also assume some familiarity with basic category theory (as in Sections 2.2-2.4 of Rune Haugseng's 2020 Lecture Notes). Ideally, you'll have taken "Rings and modules" (MA3201) and "Homological algebra" (MA3204). These two courses are not required, but recommended. We'll review these topics as needed.

If you have any questions, feel free to contact me!

## Lectures

The tentative plan is to review the fundamental group and (time permitting) discuss the following topics: covering space theory, higher homotopy groups, the long exact sequence in homotopy, cellular approximation and the Whitehead theorem, the Hurewicz theorem, Freudenthal suspension and stability, Eilenberg–MacLane spaces and Brown representability, fiber bundles and fibrations, the Serre spectral sequence, vector bundles and characteristic classes, and the Steenrod algebra.

Cal Week Lecture Date Topic Reading Exercises
2 1 Mon 09/01 Intro, review fundamental group [H 1.1, 1.2] [H 1.1.14, 1.1.16 a, b, c]
2 Fri 13/01 Siefert–van Kampen [H 1.2], [Mu2 59, 70], [Gaultieri Notes ]
3 3 Mon 16/01 Covering spaces [H 1.3], [B 3.3]
4 Fri 20/01 Path lifting, homotopy lifting property, general lifting [H 1.3], [B 3.3, 3.4] [H 1.3.7, 1.3.9]
4 5 Mon 23/01 Classification of covering spaces [H 1.3], [B 3.4, 3.5, 3.8], [Boardman Notes ] [H 1.3.14], [B 3.8.2]
6 Fri 27/01 Universal cover, higher homotopy groups [H 1.3, 4.1], [B 3.8], [Glickenstein Notes ], [M 1.1]
5 7 Mon 30/01 Properties of higher homotopy groups [H 4.1], [M 1.1] [H 4.1.2, 4.1.3]
8 Fri 03/02 Hurewicz map, relative homotopy groups and long exact sequence [H 4.1], [M 1.2] [H 4.1.6, 4.1.8]
6 9 Mon 06/02 More relative homotopy, infinite-dimensional CW-complexes, colimits [H 4.1, 0, 3.3, 3.F], [M 1.2] [H 3.3.17, 3.3.18]
10 Fri 10/02 Homotopy extension property, adjoints, cofibrations [H 0], [M 1.3], [Terilla Notes ]
7 11 Mon 13/02 HEP for CW pairs, cellular approximation theorem [H 4.1], [M 1.4], [Gutierrez cof Notes ]
12 Fri 17/02 Replacing Top, more CAT, CW approximation [H 4.1], [M 1.4, 1.8], [Gutierrez CAT Notes, CW approx Notes ]
8 13 Mon 20/02 CW approx proof, Whitehead theorem [H 4.1], [M 1.8], [Gutierrez Notes ] [H 4.1.18], [Gutierrez 10.2, 10.5, 10.8]
14 Fri 24/02 Proof of Whitehead x2, HELP [H 4.1], [M 1.8], [Gutierrez Notes ], [AGP 5.1], [Ma 10.3] [H 4.1.10, 4.1.23]
9 15 Mon 27/02 Freudenthal suspension, Hopf theorem, Hurewicz theorem [H 4.2], [M 1.5, 1.6, 1.7, 1.10], [Gutierrez Notes ]
16 Fri 03/03 Consequences of Hurewicz, Eilenberg–MacLane spaces [H 4.2], [M 1.7, 1.9, 1.10], [S Ch.6] [H 4.2.8, 4.2.9, 4.2.13, 4.2.19]
10 17 Mon 06/03 EM spaces, Dold–Thom model, loop spaces [H 4.2, 4.3], [M 1.9], [S Ch.6] [H 4.2.1]
18 Fri 10/03 Loop spaces, Brown representability [H 4.3, 4.E]
11 19 Mon 13/03 Omega spectra and cohomology theories, fiber bundles [H 4.2, 4.3, 4.E], [M 1.11]
20 Fri 17/03 LES for fiber bundle video,
Idea for spectral sequences video,
SS1 Intro and motivation video,
SS2 Graded rings and modules video,
SS3 Spectral sequence defintion video
12 21 Mon 20/03 SS4 First quadrant spectral sequence video,
SS5 Homology 1st quadrant spectral sequence video,
SS6 Unraveled exact couple (part 1) video,
SS7 Unraveled exact couple (part 2) video,
SS8 Unraveled exact couple (part 3) video,
SS9 Cohomology spectral sequence for a filtration video
[H2 Ch.5], [MT Ch.7]
22 Fri 24/03 SS10 Unraveled exact couple (part 4) video,
SS30 Double complex video,
SS31 Double complex spectral sequence video,
SS32 Some applications (part 1) video,
SS33 Some applications (part 2) video
[Vakil, Weibel, McCleary]
13 23 Mon 27/03 Fibrations [H 4.2, 4.3], [M 1.11], [RW 1.15 ], [Gutierrez Notes ]
24 Fri 31/03 Class canceled
14 Mon 03/04 No class
Fri 07/04 No class
15 Mon 10/04 No class
24 Fri 14/04 More fibrations, model categories [Dwyer and Spalinski]
16 25 Mon 17/04 More model categories, Serre spectral sequence [H2], [MT Ch. 8]
26 Fri 21/04 Serre SS computations, 1st stable stem [H2], [MT Ch. 8], [RW 2.4], [Class notes K(Z,3)] Cohom of K(Z,n) with rational coeffs via SSS
17 27 Mon 24/04 1st stable stem, stable stems are finite [Some class notes]
28 Fri 28/04 Steenrod algebra [H 4.L], [MT Ch. 3]

Wikipedia's overview of CGWH spaces as a replacement for Top.

Frankland's lecture notes on CGWH spaces.

Strickland's note on The Category of CGWH Spaces.

nLab's article on "convenient" and "nice" categories of topological spaces.

Wikipedia's article on space-filling curves.

You can read more about infinite symmetric products and the Dold–Thom theorem in [H 4.K] and [AGP 5.2].

Guillou's note on the equivariant Dold–Thom theorem recaps the classical version at the start.

Course page for previous course on spectral sequences

Vakil's note Spectral Sequences: Friend or Foe?

McCleary's book A User's Guide to Spectral Sequences

Weibel's book An Introduction to Homological Algebra

Johnson's video depicting fibers of the Hopf fibration

Wikipedia's article on the Hopf fibration

## Reference group

The members of the reference group are Denis Bondarenko Bergmann, Eivind Xu Djurhuus, and Martin Löcsei. Please feel free to pass any feedback on to them.

## Course material

There is no textbook for the course. We'll jump around a bit, but typically rely on:

## References

Other good lecture notes:

Some interesting books: