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A.Y. 2020 / 2021

First semester
Type of Learning Activity 
Programme specific subjects
Study Path 
[PDS0-2018 - Ord. 2018] common
Teaching language 


Learning objectives 

1. knowledge and understanding
The student will become familiar with the basic notions and techniques of differential topology, specifically: the notions of smooth manifold, transversality, vector bundles and cohomology
2. applying knowledge and understanding
The student will be able to use the theories and techniques taught in class for solving problems in differential topology and basic differential geometry
3. making judgements
The student will be able to recognize the features of the considered problems and decide the appropriate techniques for approaching them
4. communication skills
The student will be able to use the standard terminology and vocabulary from differential topology.
5. learning skills
The student will be able to read advanced papers and textbooks on differential topology and integrate her/his knowledge.


Basic material of linear algebra, differential calculus, general topology.


Differentiable manifolds, tangent and cotangent spaces, vector bundles. Sard's Lemma and transversality. Vector fields. Differential forms and De Rham cohomology. Integration on manifolds and Stokes Theorem. Basics of Riemannian manifolds

Teaching format 

Theoretical lessons and exercises sessions, including discussions on the exercises proposed in the lecture notes

Extended Programme 

Differentiable manifolds and differentiable functions.
Transversality and Sard's theorem.
Tangent and cotangent spaces, vector bundles.
Vector fields.
Distributions and Frobenius' theorem.
Differential forms and tensor fields.
Integration on manifolds and Stokes' theorem.
De Rham cohomology.

End-of-course test 

The final exam is aimed at testing the knowledge of the subjects of the entire course program. It is composed by a written and an oral part.

In the written exam you will be asked to solve exercises similar to those worked out in class.
The oral test will focus on theorems and proofs, a discussion of the written exam and some additional theoretical questions.

The final vote shall be determined taking into account both written and oral exams.

Other information 

Notes of the course will be available.
There will be a course web page, which will include: the dates of the exams; the notes of the course in pdf format; and other teaching material.


- John M. Lee, "Introduction to Smooth Manifolds", Springer.
- M. W. Hirsch, "Differential topology", Springer.
- J. Milnor, "Topology from the differentiable viewpoint". Princeton University Press, Princeton, NJ, 1997.
- Notes provided by the instructor

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