Knowledge and understanding: students should demonstrate to have acquired a good understanding the basics of General Topology, homotopy and the fundamental group. Applying knowledge and understanding: at the end of the course, students should be able to apply the basic techniques of topology for solving problems and exercises as well as in other areas. It is expected that students develop intuition about the most important topological spaces. Making judgements: students should demonstrate self-evaluation skills on their understanding level of the topics of the course Communication skills: students should be able to communicate, explain and present the notions and theorems learned in the course Learning skill: students should be able to integrate the topics of topology that have been studied with the topics of other courses including those of MSc in Mathematics. It is expected that students develop a good level of independence in the study that is useful to understand the proposed textbooks and for making connections with the other branches of Mathematics.

Basic knowledge of calculus, linear algebra e algebra (og the first year of the Laurea triennale.

General topology, homotopy, covering spaces, fundamental group.

1) C. Kosniowski, A First Course in Algebraic Topology, Cambridge University Press, 2009. 2) E. Sernesi, Geometria 2, Bollati Boringhieri, 2019. 3) I. M. Singer e J. A. Thorpe, Lecture Notes on Elementary Topology and Geometry, Springer-Verlag, 1967. 4) J. R. Munkres, Topology, Prentice Hall, 2000.

General and algebraic topology: topologies and topological spaces, bases, closure, interior and frontier, limit points, subspaces, product topology, continuous maps and homeomorphisms, metric spaces, countability axioms, connected and path-connected spaces, compactness and limit point compactness, Aleksandrov compactification, compact metric spaces, separability axioms (Hausdorff spaces, regular spaces and normal spaces); homotopy and path homotopy, covering spaces, path and homotopy lifting properties, fundamental group, homotopic invariance and dependence on base point, fundamental group of the circle, of the spheres, of tori and projective spaces, retractions, non-retraction theorem and Brouwer fixed point theorem, Borsuk-Ulam theorem, homotopy equivalences and deformation retractions, homotopic maps and fundamental group. Free groups, group presentations and Van Kampen theorem. Computations of fundamental groups. Covering spaces and their classification, automorphisms of a covering, regular coverings and universal covering.

Lectures and exercises. There will be a mandatory mentoring service.

Information about the course, the exams and the teaching material will be available on the Moodle page of Geometria 3 - Topologia https://moodle2.units.it Students are recommended to subscribe to the Moodle.

Six possible dates for the final exam, each consisting of a written three hours exam, and of an oral exam, focused on theory and exercises