Differential Topology
A.Y. 2021/2022
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
Since this is the first year that the course is taught by the teacher, the program is not yet defined in detail and could be updated or changed during the year. Differential topology is the study of the topology of manifolds by means of invariants defined using the differentiable structure. The methods of differential topology are vastly used in both differential and algebraic geometry. The topics of the course will be the theory of characteristic classes and Morse theory with applications ranging from the topology of complex algebraic varieties to Riemannian geometry.
In detail some of the topics covered by the course could be the following
- Morse theory and the Lefschetz theorem of hyperplane sections
- Morse coomology
- connections on vector bundles and curvature
- Thom's isomorphism, Thom class and Euler class
- Chern-Weil theory of Chern classes
- Pontrjagin classes
- Pontjagin classes and cobordism
- Stiefel-Whitney classes
- the theorem of Riemann-Roch-Hirzebruch (statement and applications)
- the Hirzebruch signature theorem (statement and applications)
- cohomology of complex algebraic hypersurfaces of the projective space
- Gauss-Bonnet theorem in even dimension
- the theorem of Hitchin-Thorpe on the obstruction to the existence of Einstein metrics
In detail some of the topics covered by the course could be the following
- Morse theory and the Lefschetz theorem of hyperplane sections
- Morse coomology
- connections on vector bundles and curvature
- Thom's isomorphism, Thom class and Euler class
- Chern-Weil theory of Chern classes
- Pontrjagin classes
- Pontjagin classes and cobordism
- Stiefel-Whitney classes
- the theorem of Riemann-Roch-Hirzebruch (statement and applications)
- the Hirzebruch signature theorem (statement and applications)
- cohomology of complex algebraic hypersurfaces of the projective space
- Gauss-Bonnet theorem in even dimension
- the theorem of Hitchin-Thorpe on the obstruction to the existence of Einstein metrics
Prerequisites for admission
We will assume some familiarity with the notions of differentiable manifold, tangent space, vector fields, differential forms, integration on manifolds, Stokes theorem. We will also assume the student will have seen the definition of de Rham cohomology with some examples of computation.
Parts of the course may overlap with topics covered in the course Algebraic Topology (e.g transversality and Morse theory), but overall the course is thought to be complementary and at the same time independent from the course Algebraic Topology.
Parts of the course may overlap with topics covered in the course Algebraic Topology (e.g transversality and Morse theory), but overall the course is thought to be complementary and at the same time independent from the course Algebraic Topology.
Teaching methods
Traditional lectures.
Teaching Resources
The lectures will be inspired by some the classical texts in differential topology. Some of these are:
- Differential topology, V. Guillemin and A. Pollack, AMS Chelsea Pubblishing
- Morse Theory, John Milnor, Princeton University Press
- Characteristic Classes, J. Milnor and J. Stasheff, Princeton University Press
- Differential Forms in Algebraic Topology, R. Bott and L. W. Tu, GTM Springer
- Differential Geometry, L. W. Tu, GTM Springer
- From Calculus to Cohomology, I. Madsen and J. Tornehave, Cambridge University Press
- Morse theory and Floer homology, M. Audin and D. Mihai, Universitext Springer
- Differential topology, V. Guillemin and A. Pollack, AMS Chelsea Pubblishing
- Morse Theory, John Milnor, Princeton University Press
- Characteristic Classes, J. Milnor and J. Stasheff, Princeton University Press
- Differential Forms in Algebraic Topology, R. Bott and L. W. Tu, GTM Springer
- Differential Geometry, L. W. Tu, GTM Springer
- From Calculus to Cohomology, I. Madsen and J. Tornehave, Cambridge University Press
- Morse theory and Floer homology, M. Audin and D. Mihai, Universitext Springer
Assessment methods and Criteria
The exam will be oral. Part of the exam will be a discussion of the exercises assigned during the course.
Professor(s)