MATH4161

**Lecturer :** Dirk Schuetz

**Term :** Epiphany 2015

- Monday 14:00 in CM 219
- Friday 14:00 in CM 219

**Problems classes:**

- Thursday 22.01, 05.02, 19.02 and 05.03 at 16:00 in CM 101

**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 | 16.01. | 26.01. | |

Problem set 2 pdf | 30.01. | 09.02. | |

Problem set 3 pdf | 09.02. | 23.02. | |

Problem set 4 pdf | 23.02. | 09.03. | |

Problem set 5 pdf | 13.03. | - |

**Links**

**Lecture Outline**

Date | Outline |

13.03. | In this lecture we give more examples of maps from a space to itself which may or may not have fixed points. |

09.03. | In this lecture we will prove the Lefschetz Fixed Point Theorem, and give plenty of examples of it. |

06.03. | In this lecture we will prove the Simplicial Approximation Theorem, the Hopf Trace Formula and define the Lefschetz number. |

02.03. | In this lecture we will have a closer look at the Simplicial Approximation Theorem. |

27.02. | In this lecture we see more duality theorems and their applications. |

23.02. | In this lecture we see a few properties of the cap-product, and meet duality theorems. |

20.02. | In this lecture we see more of the fundamental class, and we define the cap-product. |

16.02. | In this lecture we describe the fundamental class of a compact manifold with triangulation. |

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

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

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

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

30.01. | In this lecture we see how Ext makes its way into the Universal Coefficient Theorem, and what this means for singular cohomology. |

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

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

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

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

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

Last modified: 05.05.2015.