- Lecturer
- Alessandro Sisto
- Coordinator
- Davide Spriano

Classes are held in HG D7.1, on Wednesday 10-12, Friday 13-15.

* Note! The first three lectures (18, 20, 25 Sept) are held in HG E 3.*

This is an introductory course in algebraic topology, which is the study of algebraic invariants of topological spaces. Topics covered include: singular homology, cell complexes and cellular homology, the Eilenberg-Steenrod axioms.

You should know the basics of point-set topology.

Useful to have (though not absolutely necessary) basic knowledge of the fundamental group and covering spaces (at the level covered in the course "topology").

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

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

- Barycentric subdivision of a tetrahedron made by Lukas Pierce
- Video of Alexander horned sphere

Day | pages | summary |
---|---|---|

Week 1 (18-20 Sept) | Bredon 168-171, Hatcher 103-104-108 (bottom) -109 (top) | Definitions of simplices, chains, cycles, boundaries, homology Geometric intuition (Delta-complexes, orientations) |

25 Sept | Bredon 171-173 | Homology of a point, H_0(X) is determined by path-components, Hurewicz's Theorem: statement and beginning of proof |

27 Sept | Bredon 173-175 | End of proof of Hurewicz's Theorem. |

2 Oct | Bredon 176-181 | Induced maps in homology, chain complexes, chain maps, long exact sequence of homology groups (statement), homology with coefficients, relative homology, some notions of category theory. Note: category theory is not required for the exam, and was mentioned only to provide context |

9 Oct | Bredon 178-179 and 183-184 | Proof of the theorem on long exact sequences in homology (Theorem 5.6), definition of homology theory, homology of spheres (statement) |

11 Oct | Bredon 184-186 and 194-195 | Homology of spheres, reduced homology, Brouwer's fixed point theorem, definition of CW-complex |

16 Oct | Bredon 195-197 | CW-complexes: examples, compact subspaces, (very) rough outline of homology computation |

18 Oct | Bredon 200-205 | Computation of the homology of CW-complexes, definition of degree |

23 Oct | Bredon 187-188, 203-204 | Clarifications on homology of CW-complexes, basic properties of degree, reflections have degree -1. |

30 Oct | Bredon 190-192 (relevant for exam: Proposition 7.3 and Theorem 7.4) | Computation of degrees |

1 Nov | Bredon 204, 207-210, 215-216 | Relative homology of CW-complexes (no proofs), cellular approximation theorem and induced maps (no proofs), Euler characteristic |

6 Nov | Example of computation of homology of CW complexes | |

8 Nov | Bredon 219, Hatcher 111-113 | Chain homotopies, proof of the homotopy axiom |

13 Nov | Bredon, 220-224 | Cross products (no proofs), barycentric subdivisions |

15 Nov | Bredon, 224-226* *plus a more direct conclusion to the proof of excision than in Bredon. | Barycentric subdivision is chain-homotopic to the identity, subdvision operator for general chains, proof of the excision axiom. |

20 Nov | Bredon, 227, 229-230 | Homology with small simplices, Mayer-Vietoris sequence |

22 Nov | Bredon, 234-235 (see also 231-234), Dold (if you cannot access, try to connect from the ETH Wi-Fi) | Homology of complements in R^n, Jordan curve theorem and other corollaries, invariance of domain and invariance of dimension. |

27 Nov | Bredon, 233-234, 236 | Generalized Jordan curve theorem, Jordan-Brouwer separation theorem, statement of generalised Schoenflies theorem |

29 Nov | Bredon, 236-239 | Proof of the generalised Schoenflies theorem (not needed for the exam*) *but very cool! |

4 Dec | Bredon, 245-248, 250-251 | Simplicial complexes, simplicial homology, definition of simplicial map, barycentric subdivion |

6 Dec | Bredon, 251-253 | Simplicial approximation theorem |

11 Dec | Bredon, 253-256 (minus ENR discussion) | Homology with R or field coefficients, Lefschetz-Hopf fixed point theorem, sample applications (The content of this lecture is not needed for the exam). |

13 Dec | Bredon, 240-242 | Borsuk-Ulam theorem (not needed for the exam) |

18 Dec | Exercises | |

20 Dec | Final lecture: a discussion of some applications of algebraic topology (not needed for exam). |

The new exercises will be posted here.

If you have questions regarding the exercises, please don't hesitate to contact Davide Spriano.

exercise sheet | solutions | notes |
---|---|---|

Exercise sheet 1 | Solutions 1 | Path connected -> path connected component. Also, allowed arbitrary unions |

Exercise sheet 2 | Solutions 2 | |

Exercise sheet 3 | Solutions 3 | |

Exercise sheet 4 | Solutions 4 | Corrected definition of \(U\) in Question 3, Solution of Question 4 |

Exercise sheet 5 | Solutions 5 | \(j_i \colon X_i \to X\), corrected definition of splitting Question 3. |

Exercise sheet 6 | Solutions 6 | |

Exercise sheet 7 | Solutions 7 | Added bonus part Question 3. |

Exercise sheet 8 | Solutions 8; PartII | Corrected solution Ex 3. |

Exercise sheet 9 | Solutions 9; PartII | |

Exercise sheet 10 | Solutions 10; PartII | Expanded the details on computation of homology |

Exercise sheet 11 | Solutions 11 | |

Exercise sheet 12 | Solutions 12 | |

Exercise sheet 13 | Solutions 13 | |

Mock Exam | Solutions |

You can send comments/suggestions/complaints using this anonymous form.

- "Algebraic Topology", by A. Hatcher (Cambridge University Press, Cambridge, 2002). See also here.
- "Topology and geometry" by G. Bredon (Graduate Texts in Mathematics, 139. Springer-Verlag, 1997).
- "Algebraic Topology" by E. Spanier (Springer-Verlag).