- Lecturer Paul Biran
- Coordinator Patricia Dietzsch

The course will be delivered entirely online via live zoom sessions without any frontal presence. The lectures will take place

Link to zoom for the lectures:
https://ethz.zoom.us/j/92946091050?pwd=eUtmR2RRSGtTZ1JKbWt1eDYvYkkzQT09

Meeting ID: 929 4609 1050

Password: atop1HS20

Please MUTE YOUR MICROPHONES during the lectures. In case you have a question, use the zoom option to "raise your hand".

Video recordings of the sessions will be provided.

There is a forum which can be used for discussion about the course: Link to the forum.

This is an introductory course in algebraic topology. Topics covered include:

Along the way we will introduce the basics of homological algebra and category theory.You should know the basics of point-set topology.

Useful to have is a basic knowledge of the fundamental group and covering spaces (at the level usually covered in the course "topology"). Students not familiar with this topic can look this up, for example in Chapter 3, Section 1-6 and Section 8 in G. Bredon, "Topology and geometry", Graduate Texts in Mathematics, 139. Springer-Verlag, 1997. (Members of ETH can legally download the ebook trough the ETH network.)

Some knowledge of differential geometry and differential topology is useful but not necessary.

Some (elementary) group theory and algebra will also be needed.

Week | Content |
---|---|

Week 1 | Homotopy - definitions, basic constructions and properties |

Week 2 | Singular homology - definition, basic properties and first calculations |

Week 3 | Hurewicz theorem and a quick introduction to homological algebra - chain complexes and exact sequences |

Week 4 | Homology long exact sequence, relative homology and reduced homology |

Week 5 | Axiomatic approach to homology, singular homology with coefficients |

Week 6 | The homology of the sphere and related calculations, degrees of maps and applications (Brouwer fixed-point theorem) |

Week 7 | Calculation of degrees, local degrees |

Week 8 | Fundamental theorem of algebra, CW-complexes - definitions and examples |

Week 9 | Cellular homology |

Week 10 | Cellular maps, Euler characteristic and applications |

Week 11 | Homotopy axiom, cross product |

Week 12 | Excision axiom |

Week 13 | The Mayer-Vietoris long exact sequence, applications of singular homology |

Week 14 | Jordan separation theorem, invariance of domain, invariance of dimension |

If you have any questions concerning the exercises, please don't hesitate to contact Patricia Dietzsch. She will also offer regular zoom office hours, during which you can ask questions. Office hours will take place in the weeks after a new exercise sheet is posted, on Thursdays at 16:15-17:00. If possible, please ask questions in advance, via email or in the forum. You may also ask questions spontaneously during the zoom session.

Exercise sheets | Solutions | Office hours |
---|---|---|

Exercise sheet 1 | Solutions 1 | Video and pdf |

Exercise sheet 2 | Solutions 2 | Notes |

Exercise sheet 3 | Solutions 3 | Notes |

Exercise sheet 4 | Solutions 4 | Notes |

Exercise sheet 5 | Solutions 5 | Notes |

Exercise sheet 6 | Solutions 6 | Notes |

Exam | Solutions |
---|---|

Exam HS15 | Solutions HS15 |

Exam FS16 | Solutions FS16 |

Exam HS18 | Solutions HS18 |

- G. Bredon, "Topology and geometry", Graduate Texts in Mathematics, 139. Springer-Verlag, 1997. (Through the ETH Zurich network members of ETH can legally download this and other ebooks.)
- A. Hatcher, "Algebraic topology", Cambridge University Press, Cambridge, 2002. See also: A. Hatcher.
- E. Spanier, "Algebraic topology", Springer-Verlag