The readings are assigned in the textbook for this course:

Hatcher, Allen. *Algebraic Topology*. Cambridge, UK: Cambridge University Press, 2002. ISBN: 9780521795401.

The text is freely available online, but paperback copies are also available. We will be focusing on chapters 2 and 3.

WEEK # | TOPICS | READINGS |
---|---|---|

1 | CW-complexes, delta-complexes, simplicial homology, exact sequences, diagram chasing |
Chapter 0, pp. 5-8 (PDF)# Chapter 2, Section 2.1, pp. 102-108, exact sequences from ~113 (PDF) |

2 | Singular homology, homotopies and chain homotopies, categories and functors, Eilenberg-Steenrod axioms |
Chapter 2, Section 2.1, pp. 104-113 (PDF) Chapter 2, Section 2.3 (PDF) |

3 | Excision, computations for spheres, equivalence of simplicial and singular homology |
Chapter 2, Section 2.3 (PDF) Chapter 2, Section 2.1, pp. 113-134 (PDF) |

4 | Cellular homology, Mayer-Vietoris sequences, the Mayer-Vietoris argument, homology with coefficients | Chapter 2, Section 2.2, pp. 137-160 (PDF) |

5 | Tensor products, Tor, universal coefficient theorem for homology, products of simplices |
Chapter 3, Section 3.A (PDF) Chapter 3, Section 3.B (PDF) |

6 | The Eilenberg-Zilber shuffle "product" map, diagonal approximations, the Alexander-Whitney map, method of acyclic models, Kunneth formula | No readings in Hatcher text |

7 | Duality, cohomology, Ext, universal coefficients for cohomology | Chapter 3, Section 3.1, pp. 190-206 (PDF) |

8 | Projective spaces and Grassmannians, cup products, relative cup products | Chapter 3, Section 3.2, pp. 206-212 (PDF) |

9 | Dual Kunneth formula, field coefficients, cup products in cohomology of projective spaces | Chapter 3, Section 3.2, pp. 212-224 (PDF) |

10 | Manifolds, local orientations, global orientations | Chapter 3, Section 3.3, pp. 230-239 (PDF) |

11 | Cap products and choices of appropriate sign conventions, statement of Poincare duality, limits | Chapter 3, Section 3.3, pp. 239-249 (PDF) |

12 | Compactly supported cohomology, proof of Poincare duality | Chapter 3, Section 3.3, pp. 239-249 (PDF) |

13 |
Finish proof of Poincare duality Intersection pairing and cup product |
Chapter 3, Section 3.3, pp. 249-252 (PDF) |

14 | Lefschetz fixed point theorem | No readings in Hatcher text |

15 |
Finish proof of Lefschetz theorem Assorted further topics |
No readings in Hatcher text |