# Advanced Algebraic Topology

Course Code : | 2000WETAAT |

Study domain: | Mathematics |

Academic year: | 2017-2018 |

Semester: | 2nd semester |

Contact hours: | 60 |

Credits: | 6 |

Study load (hours): | 168 |

Contract restrictions: | No contract restriction |

Language of instruction: | English |

Exam period: | exam in the 2nd semester |

Lecturer(s) | Boris Shoykhet |

### 1. Prerequisites *

- competences corresponding the final attainment level of secondary school

an active knowledge of

- English

- general knowledge of the use of a PC and the Internet

general notion of the basic concepts of

The students are expected to have background in Linear Algebra, General Algebra (groups, abelian groups, rings, modules, fields), Set-theoretical topology, Calculus, and Metric Spaces.

specific prerequisites for this course

The students are expected to have at least 12 points for the !st year Master course Algebraic Topology.

### 2. Learning outcomes *

- The students will get extended knowledge on homotopy and homology groups. They will learn how to compute combinatorially the homology of a CW-complex.
- The students will learn the basic obstruction theory, including the obstructions for extension of a continuous map and of a section of a fibred bundle. They will learn the applications of this obstruction theory to the theory of characteristic classes, a theory having many applications in other mathematical fields. In particular, they will learn the Euler class and the Stiefel-Whitney classes.
- If the time permits, they will also learn the spectral sequences, and their use for computation of the cohomology groups of fibre bundles.

### 3. Course contents *

1. Computation of the singular (co)homology of a CW-complex.

2. The obstruction theory and its applications.

3. Definition of characteristic classes and their computation.

4. Characteristic classes of complex vector bundles.

5. (if the time permits) Spectral sequences and their applications.

### 4 International dimension*

The course has an international dimension.

### 5. Teaching method and planned learning activities

Personal work

### 6. Assessment method and criteria

Continuous assessment

### 7. Study material *

#### 7.1 Required reading

Allen Hatcher, Algebraic topology, in free online access at http://www.math.cornell.edu/~hatcher/AT/ATpage.html

Lecture Notes of D.B.Fuks (available at the Blackboard)

**7.2 Optional reading**

The following study material can be studied voluntarily :### 8. Contact information *

Boris Shoikhet

boris.shoikhet@uantwerpen.be