Subject: TOPOLOGIA GEOMETRICA DELLE VARIETÀ (A.A. 2020/2021)
Unit Topologia geometrica delle varietà
Related or Additional Studies (lesson)
The main purpose of the course is to provide basic tools of Algebraic and Geometric Topology with particular focus on representation theories for 3-manifolds.
Prerequisites are general topology, the concept of homotopy and fundamental group as well as the basic notions about simplicial complexes and maps and triangulations.
A list of references and supplementary material can be provided to the students that are not familiar with the above topics.
Covering spaces: lifting properties. Deck transformations. Universal covering. G-spaces and applications to covering spaces. Group actions. Existence of coverings. Monodromy of a covering.
Introduction to knot theory: definition of knot and link in the 3-sphere. Knot and link diagrams; Reidemeister moves. The fundamental group of a knot or link exterior; Wirtinger presentation.
n-dimensional manifolds and classification problems: TOP, PL and DIFF categories and their comparison in different dimensions. Basic notions of 3-manifold topology.
Representation methods for PL 3-manifolds: Heegaard splittings and diagrams. Branched coverings, Hilden-Montesinos theorem, colored links. Dehn surgery, Lickorish-Wallace theorem, Kirby diagrams.
Geometrization of 3-manifolds: the eight 3-dimensional Thurston geometries and statement of Thurston Geometrization theorem (brief outline).
Lectures including theory and exercises. The lectures will be carried out either in presence or in remote, depending on the evolution of the COVID19 situation.
Oral interview about the subjects addressed during the course or presentation of research papers about a selected topic. The exams will be carried out either in presence or in remote, depending on the evolution of the COVID19 situation.
Knowledge and understanding:
at the end of the course a student should have learned the basic features of the theory of covering spaces and basic notions and results of 3-manifold topology and 3-manifold representation theories.
Applying knowledge and understanding:
at the end of the course a student should be able to apply concepts and methods to the case of particular examples of 3-manifolds.
at the end of the course a student should be able to have a methodological approach that leads to verify claims and methods through rigorous arguments.
at the end of the course a student should be able to describe the topics presented in the course with an appropriate technical language and a correct mathematical formalism
studying the subject matter should stimulate independent learning skills and the capability of treating connected topics in further detail
- A. Hatcher, Algebraic Topology, Cambridge Univ. Press, 2002. Available online http://pi.math.cornell.edu/~hatcher/AT/ATpage.html
- A. Hatcher, Notes on Basic 3-Manifold Topology, 2007.
Available online http://www.math.cornell.edu/~hatcher/3M/3Mdownloads.html
- J. Hempel, 3-manifolds, Princeton Univ. Press, 1976.
- B. Martelli, An Introduction to Geometric Topology, CreateSpace Independent Publishing Platform, 488 pages (2016).
Available online http://people.dm.unipi.it/martelli/geometric_topology.html
- W. Massey, A basic course in Algebraic Topology, Graduate Text in Mathematics 127, Springer Verlag, 1991.
- S. Matveev – A.T. Fomenko, Algorithmic and computer methods for 3-manifolds, Mathematics and Its Applications, Springer, 2010.
- D. Rolfsen, Knots and links, American Mathematical Society, 1976.