# MA3408 Algebraic Topology 2 - Spring 2024

Schedule | Room | ||
---|---|---|---|

Lectures: | Monday | 12:15-14:00 | KJL23 |

Friday | 12:15-14:00 | KJL21 | |

Exam: | Oral exams | 15 May in KJL23 | |

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, and the Steenrod algebra.

A possible schedule is below.

Cal Week | Lecture | Date | Topic | Reading | Exercises |
---|---|---|---|---|---|

2 | 1 | Mon 08/01 | Intro, Kunneth, review fundamental group, van Kampen | [H 3.B, 1.1, 1.2] | [H 1.1.14, 1.1.16 a, b, c] |

2 | Fri 12/01 | Fundamental groupoid, covering spaces | [H 1.3], [B 3.3] | ||

3 | 3 | Mon 15/01 | Path lifting, homotopy lifting property, general lifting | [H 1.3], [B 3.3, 3.4] | [H 1.3.7, 1.3.9] |

4 | Fri 19/01 | General lifting, classification of covering spaces | [H 1.3], [B 3.4, 3.5, 3.8], [Boardman Notes ] | ||

4 | 5 | Mon 22/01 | More classification, universal cover | [H 1.3], [B 3,5, 3.8], [Boardman Notes ], [Glickenstein Notes ] | [H 1.3.14], [B 3.8.2] |

6 | Fri 26/01 | Higher homotopy groups | [H 4.1], [M 1.1] | [H 4.1.2, 4.1.3] | |

5 | Mon 29/01 | No class | |||

7 | Fri 02/02 | Action of fundamental group - video, homotopy group functor - video, covers and homotopy groups - video, products and a counterexample - video, Hurewicz homomorphism - video | [H 4.1], [M 1.1] | ||

6 | Mon 05/02 | No class | |||

Fri 09/02 | Class canceled | ||||

7 | 8 | Mon 12/02 | Relative homotopy groups and long exact sequence | [H 4.1], [M 1.2] | [H 4.1.6, 4.1.8] |

9 | Fri 16/02 | Infinite-dimensional CW-complexes, colimits, homotopy extension property | [H 0, 3.3, 3.F], [M 1.3] | [H 3.3.17, 3.3.18] | |

8 | 10 | Mon 19/02 | Adjoints, cofibrations, HEP for CW pairs | [H 4.1], [M 1.4], [Terilla Notes ], [Gutierrez cof Notes ] | |

11 | Fri 23/02 | Application of HEP, cellular approximation theorem | [H 4.1], [M 1.4, 1.8], [Gutierrez CAT Notes ] | ||

9 | 12 | Mon 26/02 | More CAT, CW approximation | [H 4.1], [M 1.8], [Gutierrez CAT Notes, CW approx Notes ] | [H 4.1.18], [Gutierrez 10.2, 10.5] |

13 | Fri 01/03 | Proof of CW approx, Whitehead theorem | [H 4.1], [M 1.8], [Gutierrez Notes ], [AGP 5.1], [Ma 10.3] | [Gutierrez 10.8] | |

10 | 14 | Mon 04/03 | Proof of Whitehead x2, HELP, Freudenthal suspension | [H 4.1, 4.2], [M 1.5, 1.6, 1.8], [Gutierrez Notes ], [AGP 5.1], [Ma 10.3] | [H 4.1.10, 4.1.23] |

15 | Fri 08/03 | Hopf theorem, stable homotopy groups, Hurewicz theorem | [H 4.2], [M 1.5, 1.6, 1.7, 1.10], [Gutierrez Notes ] | ||

11 | 16 | Mon 11/03 | Consequences of Hurewicz, Eilenberg–MacLane spaces | [H 4.2], [M 1.7, 1.9, 1.10], [S Ch.6] | [H 4.2.1, 4.2.8, 4.2.9, 4.2.13, 4.2.19] |

17 | Fri 15/03 | Loop spaces, Brown representability, omega spectra, fiber bundles | [H 4.2, 4.3, 4.E], [M 1.11] | ||

12 | 18 | Mon 18/03 | 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 | ||

19 | Fri 22/03 | 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] | ||

13 | Mon 25/03 | No class (Holiday) | |||

Fri 29/03 | No class (Holiday) | ||||

14 | Mon 01/04 | No class (Holiday) | |||

20 | Fri 05/04 | 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] | ||

15 | 21 | Mon 08/04 | Omega spectra and cohomology theories | [H 4.2, 4.3, 4.E], [M 1.11] | |

22 | Fri 12/04 | Fibrations | [H 4.2, 4.3], [M 1.11], [RW 1.15 ], [Gutierrez Notes ] | ||

16 | 23 | Mon 15/04 | More fibrations, model categories | [Dwyer and Spalinski] | |

24 | Fri 19/04 | More model categories, Serre spectral sequence | [Dwyer and Spalinski], [H2], [MT Ch. 8] | ||

17 | 25 | Mon 22/04 | Serre SS computations, 1st stable stem | [H2], [MT Ch. 8], [RW 2.4], [Class notes K(Z,3) V2023] | Cohom of K(Z,n) with rational coeffs via SSS |

26 | Fri 26/04 | 1st stable stem, stable stems are finite | [Some class notes V2023], [H2], [MT Ch. 10] | ||

18 | 27 | Mon 29/04 | Finiteness of stable stems, Steenrod algebra | [H 4.L], [MT Ch. 3] | |

28 | Fri 03/05 | Steenrod algebra, review | [MT Ch. 3, 9] |

Wikipedia's article on the comb space.

More info about the Eckmann Hilton argument.

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

Please volunteer to be part of the reference group.

## Course material

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

- [H] A. Hatcher, Algebraic Topology, Cambridge University Press, 2000.
- [M] L. Maxim, "Lecture notes on homotopy theory and applications" https://www.math.wisc.edu/~maxim/754notes.pdf

## References

Other good lecture notes:

- [Mi] H. Miller, Lectures on Algebraic Topology I from 2016
- [Mi2] H. Miller, Revised - Lectures on Algebraic Topology I from 2021
- [FH] S. Fushida-Hardy Homotopy Theory
- [RW] Oscar Randal-Williams Homotopy Theory

Some interesting books:

- [AGP] M. Aguilar, S. Gitler, C. Prieto,
*Algebraic Topology from a Homotopical Viewpoint*. Universitext Springer-Verlag, New York, 2002. - [DK] J.F. Davis, P. Kirk,
*Lecture notes in algebraic topology*. American Mathematical Society, Providence, RI, 2001. - [S] R.M. Switzer,
*Algebraic Topology - Homotopy and Homology*. Springer-Verlag GmbH Germany 2002 - [tD] T. tom Dieck,
*Algebraic topology*. European Mathematical Society (EMS), Zürich, 2008. - [B] G.E. Bredon,
*Topology and geometry*. Springer-Verlag, New York, 1993. - [DS] W.G. Dwyer, J. Spalinski,
*Homotopy theories and model categories*. Handbook of algebraic topology, 73–126, North-Holland, Amsterdam, 1995. - [MT] R.E. Mosher, M.C. Tangora,
*Cohomology Operations and Applications in Homotopy Theory*, Dover Publications, 2008. - [H2] A. Hatcher, Unpublished 5th chapter on spectral sequences.
- [MS] J. Milnor, J. Stasheff,
*Characteristic Classes*, Princeton University Press, 1974. - [Ma] J.P. May,
*A Concise Course in Algebraic Topology*, Chicago Lectures in Mathematics, 1999. - [Mu] J.R. Munkres,
*Elements of Algebraic Topology*, Westview Press, 1996. - [Mu2] J.R. Munkres,
*Topology*, Prentice-Hall, 1975. - [F] W. Fulton,
*Algebraic Topology - A First Course*, Springer-Verlag, 1995. - [V] J.W. Vick,
*Homology Theory - An Introduction to Algebraic Topology*, Springer, 1994.