# MA3408 Algebraic Topology 2: Homotopy Theory

### Web Page

The web page for this course will be set up soon, either here or in Blackboard.

To get Blackboard access, you need to

- have an active NTNU user account,
- have paid the semester fees for spring 2019, and
- have registered for the course (check at StudentWeb).

### 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 Thursdays 12-14 and Fridays 12-14 in 734, Sentralbygg II.

The first class is on 9 Jan, the last class is on 30 Apr.

### Exams

There will be oral exams with a character grade. If you would like to take the exam, please send my an email with your date and time preferences before the end of April.

### Lecture Plan

Lecture | Date | Topic | References |
---|---|---|---|

02.1 | Jan 09 | 01. Introduction and overview | notes |

02.2 | Jan 10 | Discussion and exercises | [Ghrist] Chapter 8.9 |

03.1 | Jan 16 | 02. Coverings and fiber bundles | notes |

03.2 | Jan 17 | 03. Lifting homotopies and fibrations | notes |

04.1 | Jan 23 | Discussion and exercises | |

04.2 | Jan 24 | Class cancelled | |

05.1 | Jan 30 | 04. Monodromy | notes |

05.2 | Jan 31 | Discussion and exercises | |

06.1 | Feb 06 | 05. Higher homotopy groups | notes |

06.2 | Feb 07 | 06. The long exact sequence | notes |

07.1 | Feb 13 | 07. Examples | notes |

07.2 | Feb 14 | 08. Pushouts | notes |

08.1 | Feb 20 | 09. Cofibrations | notes |

08.2 | Feb 21 | 10. Pearls and CW-complexes | notes |

09.1 | Feb 27 | Discussion and exercises | |

09.2 | Feb 28 | 11. The homology of CW complexes | notes |

10.1 | Mar 05 | Equality and pedagogical training | |

10.2 | Mar 06 | 12. The cohomology of a filtered space | notes |

11.1 | Mar 12 | 13. Spectral sequences: construction | notes |

11.2 | Mar 13 | 14. Spectral sequences: convergence | notes |

12.1 | Mar 19 | 15. The spectral sequence of a fibration | notes |

12.2 | Mar 20 | 16. Examples | notes |

13.1 | Mar 26 | 17. The Hurewicz theorem | notes |

13.2 | Mar 27 | 18. The Freudenthal theorem | notes |

14.1 | Apr 02 | 19. Eilenberg-Mac Lane spaces | notes |

14.2 | Apr 03 | 20. Killing homotopy groups | notes |

15.1 | Apr 09 | Easter break | |

15.2 | Apr 10 | Easter break | |

16.1 | Apr 16 | 21. Representation of cohomology | notes |

16.2 | Apr 17 | 22. Bockstein | notes |

17.1 | Apr 23 | 23. Unitary groups and Stiefel manifolds | notes |

17.2 | Apr 24 | 24. Grassmann manifolds and vector bundles | notes |

18.1 | Apr 30 | 25. Characteristic classes | notes |

### Textbook and References

You do not have to buy a textbook to follow this course. I will prepare notes for some parts of the class, and there will be specfic references for the others. Everyone should read Chapter 8 of [Ghrist] during the term.

[Ghrist] R. Ghrist, "Elementary Applied Topology", ed. 1.0, Createspace, 2014. https://www.math.upenn.edu/~ghrist/notes.html

[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.

[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.

### Model Categories

At some point you might want to learn about *model categories*, one of many abstraction that allows you to do homotopy theory in contexts where the objects no longer need to be topological spaces, but where you can still make sense of *fibrations*, *cofibrations*, and *equivalences*. Here are some references that are suitable introductions.

[Dwyer-Spalinski] W.G. Dwyer, J. Spaliński. Homotopy theories and model categories. Handbook of algebraic topology, 73–126. North-Holland, Amsterdam, 1995.

[Hirschhorn] P.S. Hirschhorn. The Quillen model category of topological spaces. Expo. Math. 37 (2019) 2–24.