cod. 23013

Academic year 2024/25
2° year of course - First semester
Academic discipline
Geometria (MAT/03)
Attività formative affini o integrative
Type of training activity
48 hours
of face-to-face activities
6 credits
hub: PARMA
course unit

Learning objectives

Develope a proper mathematical language.
Own abstract concept, theorems and proofs concerning differential topology.
Learn techniques to solve standard exercises and new problems, where the students need to elaborate a strategy similar to those seen in the lectures.


Basic linear algebra, differential geometry, topology, and analysis.

Course unit content

Differential topology

Full programme

Smooth manifolds:
Derivatives and tangents; fiber bundle; The inverse function Theorem; Immersions and submersion; immersions and embedding in Euclidean spaces.
Manifolds with boundary:
manifolds with boundary; derivatives and tangent spaces on manifolds with boundary; classifications of one-manifolds.
Regular and critical value:
preimage of a regular value; system of equation on a manifold; measure zero and Sard's Theorem; Whitney immersion Theorem.
transverse map to a submanifold; intersection of two submanifolds; transversality on manifolds with boundary; Brouwer fixed point Theorem; stability and generality of transversality.
Intersection theory mod 2:
intersection number mod 2; degree theory mod 2; winding number mod 2; Jordan-Brouwer separation Theorem; Borsuk-Ulam Theorem.


1. V. Guillemin, A. Pollack; Differential topology, Prentice-Hall 1974.

2. T. Brocker Guillemin, K. Janich, Introduction to Differential Topology, Cambridge University Press, 1973.

3. J.W. Milnor, Topology from the differentiable viewpoint, The University Press of Virginia, 1965.

4. A. Kosinski, Differential Manifolds, Academic Press, 1992.

Teaching methods


Assessment methods and criteria

Oral exam on exercises given during the course.

Other information

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