### Sciences

## Subject: TOPOLOGIA GEOMETRICA DELLE VARIETÀ (A.A. 2020/2021)

### master degree course in MATHEMATICS

Course year | 1 |
---|---|

CFU | 6 |

Teaching units |
Unit Topologia geometrica delle varietà
Related or Additional Studies (lesson)
- TAF: Supplementary compulsory subjects SSD: MAT/03 CFU: 6
Paola CRISTOFORI |

Exam type | oral |

Evaluation | final vote |

Teaching language | Italiano |

### Teachers

### Overview

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.

### Admission requirements

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.

### Course contents

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).

### Teaching methods

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.

### Assessment methods

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.

### Learning outcomes

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.

Making judgements:

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.

Communicating skills:

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

Learning skills:

studying the subject matter should stimulate independent learning skills and the capability of treating connected topics in further detail

### Readings

- 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.