MATH4161

**Lecturer :** Dirk Schuetz

**Term :** Epiphany 2014

- Monday 9:00am in CM 225a
- Wednesday 9:00am in CM 225a

**Literature**

- A. Hatcher, Algebraic Topology, Cambridge University Press, 2002.
- W. Fulton, Algebraic Topology: a first course, Springer Verlag, 1995.
- W.S. Massey, A basic course in Algebraic Topology, Springer Verlag, 1991.
- W.S. Massey, Singular homology theory, Springer Verlag 1980.
- E. Spanier, Algebraic Topology. McGraw-Hill, 1966.

**Assignments**

Homework | Date | Hand in | Solutions |

Problem set 1 pdf | 22.01. | 12.02. | |

Problem set 2 pdf | 05.02. | 26.02. | |

Problem set 3 pdf | 19.02. | 12.03. | |

Problem set 4 pdf | 05.03. | 30.04. | |

Problem set 5 pdf | 19.03. | - |

**Links**

**Lecture Outline**

Date | Outline |

19.03. | In this lecture we look we prove Alexander duality for spheres embedded in spheres, and use this to prove the generalized Jordan Curve Theorem. |

17.03. | In this lecture we look at the direct limit without calling it the direct limit, and use this to prove special cases of Alexander duality. |

12.03. | In this lecture we see the Alexander and Lefschetz Duality Theorems, and some applications of them. |

10.03. | In this lecture we see have a problems, problems, Problems Class. |

05.03. | In this lecture we see some properties of the cap product, and state the Poincaré Duality Theorem. |

03.03. | In this lecture we describe the fundamental class of a compact manifold with triangulation and the cap-product. |

26.02. | In this lecture we define homology with coefficients and look at the homology of manifolds. |

24.02. | In this lecture we discuss naturality of the cup product, and show how it can be used to calculate more cohomology rings. |

19.02. | In this lecture we calculate the cohomology ring of the torus, discuss commutativity of the cup product and give relative versions of it. |

17.02. | In this lecture we show some calculations of the cup product, especially for the torus and the projective plane. |

12.02. | In this lecture we define the cup-product, and show how it turns cohomology into a ring. |

05.02. | In this lecture we see applications of the Universal Coefficient Theorem, and use it to calculate cohomology groups of certain spaces. |

03.02. | In this lecture we see some more properties of Ext, and meet the Universal Coefficient Theorem. |

29.01. | In this lecture we define Ext, the right derived friend of Hom, and give some calculations. |

27.01. | In this lecture we obtain the long exact sequences in cohomology of a pair and of a union of two open sets. |

22.01. | In this lecture we define cochain complexes and the singular cohomology groups of a topological space. We will also consider certain properties and examples. |

20.01. | In this lecture we consider the set of homomorphisms between two abelian groups and study some of its properties. |

Last modified: 09.05.2014.