# 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.

#### Exam information

Here you will find some practical information about the exam.

Here you will some notes from the revision lecture.

Here you will find the preliminary exam schedule.

### 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 12-14 in EL4 and Fridays 10-12 in B3.

The first class is on 12 Jan, the last class is on 7 May.

**Important (updated 03/01/2020): As per instructions from the Rector, (at least) for the first week we will begin with online lectures.**

**Zoom link: https://ntnu.zoom.us/j/91971337094?pwd=bWhDdk83b0ZSWmUzcGJVZ3dmSGRKQT09 **

### 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.

### Reference group

The reference group consists of Elias, Thomas, and Benjamin.

### Lecture Plan

Here is a preliminary lecture plan. This will be updated as the course goes on (it may be overly optimistic).

Semester Week | Topics | Exercises | Solutions | Videos | Notes | References |
---|---|---|---|---|---|---|

Week 2 | Higher homotopy groups and relative homotopy groups | Week 1 | Week 1 | Week 1 | [Maxim, 1.1 and 1.2] | |

Week 3 | Fibrations, cofibrations and the cellular approximation theorem | Week 2 | week2_sol.pdf | week_2.pdf | [Maxim, 1.3, 1.4, 1.11] | |

Week 4 | Whitehead theorem, excision and the Freudenthal suspension theorem. | week3.pdf | week3_sol.pdf | Week 3 Lecture 2 | week_3.pdf | [Maxim, 1.5, 1.6, 1.7] |

Week 5 | CW Appromxation, Eilenberg–MacLane spaces | week4.pdf | week4_sols.pdf | Week 4 Lecture 1. Week 4 Lecture 2 (first half only, sorry). | week_4.pdf | [Maxim, 1.8, 1.9] |

Week 6 | Hurewicz theorem, more about fibrations and fiber bundles | week5.pdf | week5_sol.pdf | Week 5 Lecture 1. Week 5 Lecture 2. | week_5.pdf | [Maxim, 1.10, 1.12,1.13] |

Week 7 | Introduction to spectral sequences | week6.pdf | week6_sol.pdf | Week 6 Lecture 1 Week 6 Lecture 2 Sorry, I once again forgot to record the second half of the lecture. But the example came from pages 68 - 70 here. | week_6.pdf | [Maxim, 2.1 and 2.2] |

Week 8 | Serre spectral sequence | week7.pdf | week7_sol.pdf | Week 7 Lecture 1 Week 7 Lecture 2 | week_7.pdf | [Maxim, 2.3 and 2.4] |

Week 9 | Gysin and Wang sequence, mod C theory, exact couples | week8.pdf | week_8_sol.pdf | Week 8 Lecture 1 Week 8 Lecture 2 | week_8.pdf | [Maxim, 2.5,2.6,2.7] |

Week 10 | Fiber bundles and principal bundles | Week 9 Lecture 1 | week_9.pdf | [Maxim, 3.1 and 3.2] | ||

Week 11 | Classification of principal G-bundles | week10.pdf | week10_sols.pdf | Week 10 Lecture 1 Week 10 Lecture 2 | week_10.pdf | [Maxim, 3.3] |

Week 12 | Chern classes and Steifel–Whitney classes | week11.pdf | week11_sol.pdf | Week 11 Lecture 1 Week 11 Lecture 2 | week_11.pdf | [Maxim, 4.1 and 4.2] |

Week 13 | Easter Break | |||||

Week 14 | Orientations and Pontryaigin classes | Week 12 Lecture 1 | week_12.pdf | [Maxim, 4.3 and 4.4] | ||

Week 15 | K-theory part 1 | week13.pdf | Week 13 Lecture 1 Week 13 Lecture 2 | week_13.pdf | Hatcher, Section 2.1 | |

Week 16 | K-theory part 2 | week14.pdf | week13_sol.pdf | Week 14 Lecture 1 Week 14 Lecture 2 | week_14.pdf | |

Week 17 | Revision | week14_sol.pdf |

### 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:

[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.