Differential Topology
A.Y. 2025/2026
Learning objectives
The aim of the course is to illustrate the main results and to provide some of the techniques of differential topology.
Expected learning outcomes
Know how to use some of the differential topology techniques on differentiable manifolds.
Lesson period: Second 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
Second semester
Course syllabus
Recalls on: differentiable varieties, differentiable applications, immersions and submersions.
Critical/regular points, critical/regular values, and differentiable structure of the pre-image.
Transversality and stability properties.
Sard's theorem and Whitney's theorems.
Manifolds with boundary. Thom's Transversality Theorem. Tubular neighborhood. Genericity and Extension Theorem.
Oriented manifolds, orientation of the pre-image, and orientation numbers.
Intersection numbers. Degree of a map.
Recalls on vector bundles, vector fields. Poincaré-Hopf theorem.
Hopf's degree theorem. Integral of a k-form and the degree formula.
Recalls on singular homology. Cellular homology.
Morse functions. Index of a non-degenerate critical point. Morse lemma. First and second Morse theorems.
Topology of the sublevel manifolds. Reeb's theorem.
Morse inequalities and equalities.
Applications to the classification of 1- and 2-dimensional manifolds.
Lefschetz theorem for hyperplane sections.
Critical/regular points, critical/regular values, and differentiable structure of the pre-image.
Transversality and stability properties.
Sard's theorem and Whitney's theorems.
Manifolds with boundary. Thom's Transversality Theorem. Tubular neighborhood. Genericity and Extension Theorem.
Oriented manifolds, orientation of the pre-image, and orientation numbers.
Intersection numbers. Degree of a map.
Recalls on vector bundles, vector fields. Poincaré-Hopf theorem.
Hopf's degree theorem. Integral of a k-form and the degree formula.
Recalls on singular homology. Cellular homology.
Morse functions. Index of a non-degenerate critical point. Morse lemma. First and second Morse theorems.
Topology of the sublevel manifolds. Reeb's theorem.
Morse inequalities and equalities.
Applications to the classification of 1- and 2-dimensional manifolds.
Lefschetz theorem for hyperplane sections.
Prerequisites for admission
We assume that the students have basic knowledge about general topology, fundamental group, differentiable manifolds and differential maps.
Teaching methods
Lectures.
Teaching Resources
M. Abate, F. Tovena, Geometria Differenziale, New York Springer-Verlag 2011.
V.Guillemin, A. Pollack Differential Topology. AMS Chelsea Publishing 2010.
J.W. Milnor, Topology from the Differentiable Point of View. University press of Virginia.
J.W. Milnor, Morse theory, Princeton University Press
V.Guillemin, A. Pollack Differential Topology. AMS Chelsea Publishing 2010.
J.W. Milnor, Topology from the Differentiable Point of View. University press of Virginia.
J.W. Milnor, Morse theory, Princeton University Press
Assessment methods and Criteria
The final examination consists of an oral exam. In the oral exam, the student will be required 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.
MAT/03 - GEOMETRY - University credits: 6
Lessons: 42 hours
Professors:
Bertolini Marina, Colombo Elisabetta
Professor(s)
Reception:
friday.8.45-11.45
Office2101, second floor, via C. Saldini 50