# Algebraic topology

Course Code : | 2000WETATO |

Study domain: | Mathematics |

Academic year: | 2017-2018 |

Semester: | 1st semester |

Contact hours: | 60 |

Credits: | 6 |

Study load (hours): | 168 |

Contract restrictions: | No contract restriction |

Language of instruction: | English |

Exam period: | exam in the 1st 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

specific prerequisites for this course

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

### 2. Learning outcomes *

- The students will get knowledge on some basic homotopy invariants one assigns to a topological space, such as: the fundamental group, the homology groups, and the higher homotopy groups.
- The students will learn how to compute (some of) homotopy type invariants of topological spaces.
- They will also learn how to apply these invariants for solution of some problems, e.g.: how to prove that any continuous map on n-dimensional ball to itself has a fixed point? e.g. how to prove that the spheres S^m and S^n are not homotopy equivalent for m not equal to n?

### 3. Course contents *

1. General notion of a homotopy invariant of topological spaces.

2. CW-complexes.

3. The coverings and the fundamental group; examples of computation; Van Kampen theorem.

4. The higher homotopy groups: easy to define but hard (in general impossible) to compute.

5. The homology and cohomology groups: harder to define but computable.

6. Computation of H(S^n), degree of a map.

7. The Euler characteristic of a triangulated space, independence on triangulation, the Lefschetz fixed-pont theorem, Hopf theorem on zeroes on a vector field on a manifold.

### 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

1. Several chapters of the Allen Hatcher's book Algebraic Topology, in free access via

http://www.math.cornell.edu/~hatcher/AT/ATpage.html

2. Lectures 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