Geometry 5
A.Y. 2026/2027
Learning objectives
The aim of the course is to give elements of Covering Theory and of De Rham cohomology.
Expected learning outcomes
To be able to recognize a covering space and its basic properties. To be able to classify the coverings of a given topological space via its fundamental group.
To be able to compute de Rham cohomology of simple differentiable manifolds.
To be able to compute de Rham cohomology of simple differentiable manifolds.
Lesson period: First semester
Assessment methods: Esame
Assessment result: voto verbalizzato in trentesimi
Single course
This course cannot be attended as a single course. Please check our list of single courses to find the ones available for enrolment.
Course syllabus and organization
Single session
Responsible
Lesson period
First semester
Course syllabus
Tha course has two versions: 6 cfu version or 9 cfu version.
The program for the 6 cfu version is:
CW complexes and classification of compact topological surfaces
Topological coverings and their properties.
Monodromy. Universal covering. ClassificationTheorem.
Differentiable coverings and orientation covering.
Homological Algebra.
de Rham complex and cohomology. Mayer-Vietoris sequence. Poincaré lemma. Finiteness theorems.
Complements of differential geometry and algebraic topology.
The program for the 9 cfu version contains all the topics of the 6 cfu version and:
Poincaré lemma. Cohomology with compact supports, Poincaré duality.
Singular homology groups, Mayer Vietoris exact sequence. Computation of homology groups in some examples. de Rahm Theorem.
Complements of differential geometry and algebraic topology.
The program for the 6 cfu version is:
CW complexes and classification of compact topological surfaces
Topological coverings and their properties.
Monodromy. Universal covering. ClassificationTheorem.
Differentiable coverings and orientation covering.
Homological Algebra.
de Rham complex and cohomology. Mayer-Vietoris sequence. Poincaré lemma. Finiteness theorems.
Complements of differential geometry and algebraic topology.
The program for the 9 cfu version contains all the topics of the 6 cfu version and:
Poincaré lemma. Cohomology with compact supports, Poincaré duality.
Singular homology groups, Mayer Vietoris exact sequence. Computation of homology groups in some examples. de Rahm Theorem.
Complements of differential geometry and algebraic topology.
Prerequisites for admission
We assume that the students have basic knowledge about topology, fundamental group, differentiable manifolds and differential forms.
Teaching methods
Lectures and exercise classes.
Teaching Resources
For the 6 cfu version:
M Manetti, Topologia, Springer, 2008
A. Hatcher, Algebraic Topology, Cambridge Univ. Press, 2002 (https://pi.math.cornell.edu/~hatcher/AT/AT.pdf) R. Bott, L. Tu, Differential Forms in Algebraic Topology. New York Springer Verlag 1982
M.Abate, F. Tovena, Geometria Differenziale, New York Springer-Verlag 2011
For the 9 cfu version, all the above references and:
R. Bott, L. Tu, Differential Forms in Algebraic Topology. New York Springer Verlag 1982
M Manetti, Topologia, Springer, 2008
A. Hatcher, Algebraic Topology, Cambridge Univ. Press, 2002 (https://pi.math.cornell.edu/~hatcher/AT/AT.pdf) R. Bott, L. Tu, Differential Forms in Algebraic Topology. New York Springer Verlag 1982
M.Abate, F. Tovena, Geometria Differenziale, New York Springer-Verlag 2011
For the 9 cfu version, all the above references and:
R. Bott, L. Tu, Differential Forms in Algebraic Topology. New York Springer Verlag 1982
Assessment methods and Criteria
The final examination consists of an oral exam. In the oral exam, the student will be required to solve some exercises and to illustrate results presented during the course, in order to evaluate her/his knowledge and comprehension of the arguments covered as well as the capacity to apply them.
Final marks are given using the numerical range 0-30, and will be communicated immediately after the oral examination.
Final marks are given using the numerical range 0-30, and will be communicated immediately after the oral examination.
MATH-02/B - Geometry - University credits: 9
Exercises: 36 hours
Lessons: 42 hours
Lessons: 42 hours
Professors:
Bertolini Marina, Matessi Diego
Professor(s)