# MA3408 Algebraic Topology 2: Homotopy Theory

### Course Content

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.

### 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 somewhat familiar with homology and cohomology. It's also good if you have some familiarity with fundamental groups and covering spaces, but I will recall the definitions in places where I need (to generalize) them.

### Class Hours

Classes are Tuesdays 2-4 and Fridays 10-12 both in MA23. We will begin with (at least) two weeks of digital lectures. When we eventually move to in-person lectures, I will record the lectures (if possible).

The Zoom link for the beginning of the semester is: https://NTNU.zoom.us/j/98885329051?pwd=QmNKbWpNait3eHk4L3FrUWc5OEVKZz09

### Discourse

I have set up a Discourse server for the class here. This is a message board where you can post questions (even anonymously) that you have about the course topics. There is a new user guide for Discourse if you have never used it before: https://meta.discourse.org/t/discourse-new-user-guide/96331 and the NTNU help page is https://wiki.math.ntnu.no/drift/help/forum.

### Exams

There will be oral exams with a character grade. Exam timetable (updated 09/05)

### Lectures

Lecture | Date | Topic | Notes | Videos | Reference | Exercises | Solutions |
---|---|---|---|---|---|---|---|

1 | 11/01 | Review of homotopy, higher homotopy groups | Lecture 1 | Lecture 1 | [Maxim, Section 1.1] | Week 1 Exercises | Week 1 solutions |

2 | 14/01 | Relative homotopy groups, long exact sequence in homotopy | Lecture 2 | Lecture 2 | [Maxim, Section 1.2] | ||

3 | 18/01 | Homotopy extension property, cofibrations | Lecture 3 | Lecture 3 | [AGP, Section 4.1, 4.2] | Week 2 exercises | Week 2 solutions |

4 | 21/01 | Homotopy lifting property, fibrations | Lecture 4 | Lecture 4 | [AGP, Section 4.3, 4.4] | ||

5 | 25/01 | Homotopy extension and lifting property, Whitehead theorem | Lecture 5 | Lecture 5 | [May, Chapter 10] | Week 3 exercises | Week 3 solutions |

6 | 28/01 | Cellular approximation, admissible topologies | Lecture 6 | Lecture 6 | [May, Chapter 10.4] | ||

7 | 01/02 | Homotopy excision, Freudenthal suspension | Lecture 7 | Unavailable due to technical error! | [Maxim, Section 1.5-1.6] | Week 4 exercises | Week 4 solutions |

8 | 04/02 | CW-approximation | Lecture 8 | Lecture 8 (now with sound!) | [Maxim, Section 1.8] | ||

9 | 08/02 | Eilenberg–MacLane spaces, Hurewicz theorem | Lecture 9 | Lecture 9 | [Maxim, Section 1.9,1.10] | Week 5 exercises | Week 5 solutions |

10 | 11/02 | Brown representability | Lecture 10 | Lecture 10 | |||

11 | 15/01 | Filtered complexes, spectral sequences | Lecture 11 | Lecture 11 | Week 6 exercises | Week 6 solutions | |

12 | 18/02 | Double complexes, Serre spectral sequence | Lecture 12 | Lecture 12 | |||

13 | 22/02 | Serre spectral sequence - examples | Lecture 13 | Lecture 13 | Week 7 exercises | Week 7 solution | |

14 | 25/02 | Serre spectral sequence in cohomology | Lecture 14 | Lecture 14 | |||

15 | 01/03 | Gysin and Wang sequences, Serre classes | Lecture 15 | Lecture 15 | [Maxim, Section 2.4,2.5] | Week 8 exercises | Week 8 solutions |

16 | 04/03 | Odds and ends on spectral sequences | Lecture 16 | Lecture 16 | |||

17 | 08/03 | Fiber bundles - first definition and examples | Lecture 17 | Lecture 17 | Week 9 exercises | Week 9 solutions | |

18 | 11/03 | Principal G-bundles | Lecture 18 | Lecture 18 | |||

19 | 15/03 | Morphisms of bundles | Lecture 19 | Lecture 19 | [Maxim, Section 3.1,3.2] | Week 10 exercises | Week 10 solutions |

20 | 18/03 | Pullback along homotopic maps | Lecture 20 | Lecture 20 | [Maxim, Section 3.2] | ||

21 | 22/03 | Classifying spaces, homotopy classification of bundles | Lecture 21 | Lecture 21 | [Maxim, Section 3.3] | Week 11 exercises | Week 11 solutions |

22 | 25/03 | Review of bundles | Lecture 22 | Lecture 22 | |||

23 | 29/03 | Chern classes for complex vector bundles | Lecture 23 | Lecture 23 | [Maxim, Section 4.1] | Week 12 exercises [now updated, April 1st] | Week 12 solutions |

24 | 01/04 | Stiefel–Whitney classes for real vector bundles | Lecture 24 | Lecture 24 | [Maxim, Section 4.2] | ||

25 | 05/04 | Applications of characteristic classes | Lecture 25 | Lecture 25 | [Maxim, Section 4.3] | Week 13 exercises | Week 13 solutions |

26 | 08/04 | Pontryagin classes | Lecture 26 | Lecture 26 | [Maxim, Section 4.4] | ||

27 | 19/04 | Exam revision/review lecture | Revision lecture | Revision lecture | Review preview weeks exercises! | ||

28 | 22/04 | Office hours |

### Textbook and References

You do not have to buy a textbook to follow this course. We will roughly follow the notes by Maxim below.

[Maxim] L. Maxim, "Lecture notes on homotopy theory and applications" https://www.math.wisc.edu/~maxim/754notes.pdf

Other useful references are:

[AGP] Algebraic Topology from a Homotopical Viewpoint, Universitext book series

[Arkowitz] Introduction to homotopy theory, Spring-Verlag, 2011

[Bott-Tu] R. Bott, L.W. Tu. Differential forms in algebraic topology. Springer-Verlag, New York-Berlin, 1982.

[Bredon] G.E. Bredon. Topology and geometry. Springer-Verlag, New York, 1993.

[Davis-Kirk] J.F. Davis, P. Kirk. Lecture notes in algebraic topology. American Mathematical Society, Providence, RI, 2001.

[Hatcher] Hatcher, A. Algebraic topology. https://pi.math.cornell.edu/~hatcher/AT/AT.pdf

[tom Dieck] T. tom Dieck. Algebraic topology. European Mathematical Society (EMS), Zürich, 2008.

[May] J.P. May. A concise course in algebraic topology. University of Chicago Press, Chicago, IL, 1999.