TOPICS IN ADVANCED GEOMETRY

Degree course: 
Corso di First cycle degree in MATHEMATICS
Academic year when starting the degree: 
2015/2016
Year: 
3
Academic year in which the course will be held: 
2017/2018
Course type: 
Compulsory subjects, characteristic of the class
Credits: 
8
Period: 
Second semester
Standard lectures hours: 
64
Detail of lecture’s hours: 
Lesson (64 hours)
Requirements: 

Linear Algebra and Geometry, Algebra 1, Geometry 1, Algebra 2.

Final Examination: 
Orale

In accordance with the Course aims, the verification of the learning skills will derive from:
- the results obtained in the exercise sheets during the course;
- a written exam, where the student will prove his ability to compute the fundamental and Homology groups and the possible coverings in concrete cases.
- an oral exam, during which the student should prove to have understood and be able to reproduce the proofs of the most relevant theorems treated in the course.

Assessment: 
Voto Finale

The course is an introduction to Algebraic Topology. The aim is that the student learns and can apply the basic techniques in Algebraic Topology, such as the computation of fundamental group and of homology groups of a CW complex, and the study of the coverings of a topological space.

1) Complements on the fundamental group: Van Kampen's Theorem and other computation methods.

2) Covering Theory: coverings, lifting properties. Galois classification of coverings. Monodromy of coverings.

3) Classification of Topological surfaces: Jordan's curve Theorem; connected sum, Euler-Poincaré characteristic, orientability. Classification.

4) Homology theory, first definitions: CW complexes and simplicial complexes. Simplicial homology theory. Homology of surfaces. Relative homology.

5) Homology theory, first properties and techniques: Simplicial approximation. Exact sequences and excision. Applications. Cellular homology. Mayer-Vietoris sequence. Relation with the fundamental group.

1) C. Kosniowski – A first course in Algebraic Topology, Cambridge University Press.
2) A. Hatcher - Algebraic Topology, Cambridge University Press (free on the web)
3) J. Munkres, Elements of Algebraic Topology, Addison-Wesley.

Frontal lessons. Exercise sheets to be solved at home, that will be periodically discussed in class.

Office hours by appointment: send me an email.

Borrowed from

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