Geometry 5
A.Y. 2020/2021
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
Teaching methods
Teaching activity will be held following current regulations.
All the lectures and exercise sessions will take place on Zoom and the students will be able both to attend in real time as established in the schedule of the first semester, and/or to watch the video later, since all the sessions will be recorded and made available for the students on Ariel.
Program and references
The program and references for the course will not change.
Examination
The oral part of the examination will be at the university or on Zoom platform depending on current regulations.
Teaching activity will be held following current regulations.
All the lectures and exercise sessions will take place on Zoom and the students will be able both to attend in real time as established in the schedule of the first semester, and/or to watch the video later, since all the sessions will be recorded and made available for the students on Ariel.
Program and references
The program and references for the course will not change.
Examination
The oral part of the examination will be at the university or on Zoom platform depending on current regulations.
Prerequisites for admission
We assume that the students have basic knowledge about general 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.
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.
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
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.
Hints on homology and on de Rham Theorem.
Complements of differential geometry and algebraic topology.
Poincaré lemma. Cohomology with compact supports, Poincaré duality.
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
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
Modules or teaching units
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