Algebraic topology

Lecturer: Roland van der Veen, Assistants: Jorge Becerra and Sjabbo Schaveling. Contact us at: algebraictopology18@gmail.com

Lectures: 14-17, HFG 611AB (or MIN 201, weeks 38,41,44,45, In week 42 Ruppert D(80), In week 43 BBG 061 (64)).

Virtual Office Hours (experimental! starting sept 21) Fridays 14-15

**Aim of the course**
This course is an introduction to Algebraic Topology. Its main topic is the study of homology groups of topological spaces. These homology groups provide algebraic invariants of topological spaces which can be computed in many examples of interest.
In the first part of the course we will construct the singular homology groups of topological spaces and establish their basic properties, such as homotopy invariance and long exact sequences. In the second part of the course we will introduce CW-complexes. These provide a useful class of topological spaces with favorable properties, and we will explain how the homology of CW-complexes can be computed using cellular homology. We will also discuss some basic concepts from homotopy theory.

**Final grade**
The final grade is mostly determined by the final exam on Wed Jan 23 14:00-16:30 KBG COSMOS, Utrecht. The exam is a written exam on all material treated in the course. You are not allowed to use notes.
The regular homework assignments can either be handed in by email or in person before class. Homework is optional can only increase your final grade. More formally: $\mathrm{FinalGrade} =
\min\{\mathrm{rank}(\pi_3(S^2))\ \mathrm{ExamGrade}+\sin(\frac{\pi}{20}\mathrm{Average Homework}),10\}$.

The retake exam is on Wed Feb 27, 14-17 in Ruppert 033. It is optional and may be used to replace your exam grade in the above formula, if your home institution agrees.

**Prerequisites**
- Background in point-set topology: topological spaces, continuous maps, compactness, quotients and products, along the lines of these notes by A. Hatcher and maybe a first encounter with the fundamental group.
- Knowledge about basic constructions with vector spaces and abelian groups.
- Some familiarity with categories and functors is also helpful. For those who haven't seen this before, the "Intensive course on Categories and Modules" is recommended.

LITERATURE: Notes by Steffen Sagave. A few corrections/typos, please tell me if you find more!

EXERCISES: Exercises by Steffen Sagave

VIDEO RECORDINGS (Thanks to Mike Daas):

Recordings of all the lectures will be available here, you'll need the password: Xz4F

$\pi_n(X,A,x_0)\cong H_n(X,A,\mathbb{Z})$

$\cdots \to H_n(\bar{C}) \to H_{n-1}(C')\to H_{n-1}(C)\to \cdots$

$\pi_3(S^2) = \mathbb{Z}$

$\partial_n = \sum_{i=0}^n (-1)^i d_i:C_n\to C_{n-1}$

SCHEDULE

Date | Class material | Exercises |
---|---|---|

Wed Sep 12 | Introduction, Lecture 1 | Homework due Sept 19 before class |

Wed Sep 19 in MIN 201 | Lecture 3, homology of spheres | Homework due Sept 26 before class: Exercise 1 from Exercises 19-9 and Exercise 2.4 from Exercises 2016 by Steffen Sagave. |

Wed Sep 26 | Lecture 3, 4 more on Long Exact Sequence, proof of Homotopy invariance (without simplicial sets) | Exercises 26-9, Exercises 1,2 |

Wed Oct 3 | Lecture 6 proof of small simplices theorem (Following Hatcher p.119 prop 2.21) | Homework due Oct 10 before class:Exercises 1,2 from Exercises 3-10 |

Wed Oct 10 | Lecture 5, proof of Excision theorem, Degree of a map, lect 7 cell attachment | Exercise 1 from Exercises 10-10 and Exercise 7.4 from Exercises 2016 by Steffen Sagave |

Wed Oct 17 (in Ruppert D) | Lecture 7, Hawaiian earring | Exercises 1,2 from Exercises 17-10 |

Wed Oct 24 (in BBG 061) | Lecture 8 CW complexes | Exercises 8.1, 8.4 from Exercises 2016 by Steffen Sagave. It may be helpful to also check out Jorge's comments on torsion for 8.4 |

Wed Oct 31 (in Minnaert 201) | Lecture 9, Cellular homology | No homework. |

Wed Nov 7 | Lecture 10 | Homework, Sagave Exercise 10.1 and this exercise. |

Wed Nov 14 | Lecture 10 | Homework: this exercise by Jorge |

Wed Nov 21 | Lecture 11 | Homework: Sagave Exercise 11.4 |

Wed Nov 28 | Lecture 12 | Homework: Sagave Exercise 12.1 |

Wed Dec 5 | Lecture 13 | |

Wed Dec 12 | Lecture 14 | Practice exam by Jorge |

Wed Jan 23 | EXAM 14:00-17:00 KBG COSMOS | The exam is a written exam on all material treated in the course. You are not allowed to use notes. Solutions to the exam |

Wed Feb 27 | RETAKE EXAM 14:00-17:00 Ruppert 033 | The exam is a written exam on all material treated in the course. You are not allowed to use notes. |

LINKS

**Solutions to selected exercises** Solutions to Selected exercises by Jorge.
Solutions to Sagave's exercises week 4.

**Additional exercises** Additional practice exercises by Jorge.
Additional practice exercises Lecture 9 by Jorge.

**The Hawaiian earring** has a non-free fundamental group, see an old paper by Bart de Smit (Leiden).

**Computational homology** is a vibrant area of topological data analysis

**Finite topological spaces** are surprisingly rich and complicated. My favourite is the pseudocircle
a four element set that has fundamental group $\mathbb{Z}$. There are 9 non-equivalent topologies on the 3-element set. As the number of elements grows, so does the number of topologies.

**The higher homotopy groups of spheres** $\pi_n(S^m)$ defined in Lecture 13 of the notes are some of the weirdest and perhaps deepest features of algebraic topology.
Roughly speaking they are Abelian groups measuring in how many distinct ways an n sphere fits into an m sphere. The answer is a baffling mix of order and chaos. I like $\pi_{10}(S^4) = \mathbb{Z}/3\mathbb{Z} \times \mathbb{Z}/24\mathbb{Z}$. Much remains to be done!

**Lecture 1** Categories and functors [Steffen Sagave]

1.1 Categories, examples,

1.2 more examples,

1.3 subcategories, epimorphisms and monomorphims,

1.4 functors,

1.5 natural transformations, equivalences.

**Lecture 2** The category of R-modules [Bas Edixhoven and Arno Kret]

2.1 left modules, right modules, examples,

2.2 another definition, initial objects,

2.3 morphisms, the categories R-mod and mod-R,

2.4 morphisms between free modules, k[x]-mod,

2.5 submodules, quotients,

2.6 products and sums,

2.7 continuation of 2.6,

2.8 finitely generated modules over a principal ideal domain,

2.9 classification, application to finitely generated Abelian groups and Jordan normal form.

**Lecture 3** Tensor products, and exact sequences [Rob de Jeu]

Both topics rely heavily on knowledge of R-modules, and to a lesser extent on knowledge of functors. The part on exact sequences is mostly independent of the part on tensor products, but tensor products show up in some of the examples here.

3.1 tensor products: definitions

3.2 basic properties of tensor products

3.3 exact sequences, part 1 (definitions, splitness, retractions)

3.4 exact sequences, part 2 (exact, left exact, and right exact functors) (one more example will be added to the end of this video in the next few weeks)

**Lecture 4** Yoneda, Limits and Adjunction [David Holmes]

Typed notes for this section are available, see the `resources' at the bottom of the page.

4.1 Yoneda's Lemma; see also this extra example

4.2 Initial and Terminal objects (recap)

4.3 Limits

4.4 Colimits

4.5 More examples

4.6 Adjunction

Remark: 4.1 may be somewhat confusing on a first viewing. Sections 4.2-4.5 are rather simpler, and are independent of 4.1. Notes for Lecture 4.

- Glen E. Bredon, Topology and Geometry
- Allen Hatcher, Algebraic Topology
- Tammo tom Dieck, Algebraic Topology
- For more point-set topology background: in addition to Hatcher's notes the book Topology by K. Jaenich is recommended.