Geometry 5

A.Y. 2021/2022
9
Max ECTS
78
Overall hours
SSD
MAT/03
Language
Italian
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.
Course syllabus and organization

Single session

Responsible
Lesson period
First semester
More specific information on the delivery modes of training activities for academic year 2021/22 will be provided over the coming months, based on the evolution of the public health situation.
Prerequisites for admission
We assume that the students have basic knowledge about topology, differentiable manifolds and differential forms.
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.
Geometry 5 (first part)
Course syllabus
CW complexes and classification of compact topological surfaces
Topological coverings and their properties.
Monodromy. Universal covering. ClassificationTheorem.
Homological Algebra.
de Rham complex and cohomology. Mayer-Vietoris sequence. Poincaré lemma. Finiteness theorems.
Complements of differential geometry and algebraic topology.
Teaching methods
Lectures, exercise classes and guided exercises.
Teaching Resources
M Manetti, Topologia, Springer, 2008
A. Hatcher, Algebraic Topology, Cambridge Univ. Press, 2002 (http://www.math.cornell.edu/~hatcher/AT/ATpage.html)
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
Geometry 5 mod/2
Course syllabus
All the topics of the 6 cfu version and:
Poincaré lemma. Cohomology with compact supports, Poincaré duality.
Differentiable coverings and orientation covering.
Hints on homology and on de Rham Theorem.
Complements of differential geometry and algebraic topology.
Teaching methods
Lectures, exercise classes and guided exercises.
Teaching Resources
M Manetti, Topologia, Springer, 2008
A. Hatcher, Algebraic Topology, Cambridge Univ. Press, 2002 (http://www.math.cornell.edu/~hatcher/AT/ATpage.html)
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
Geometry 5 (first part)
MAT/03 - GEOMETRY - University credits: 6
Lessons: 42 hours
Professor: Colombo Elisabetta
Geometry 5 mod/2
MAT/03 - GEOMETRY - University credits: 3
Practicals: 36 hours
Professor: Camere Chiara
Professor(s)
Reception:
friday.8.45-11.45
Office2101, second floor, via C. Saldini 50