GEOMETRY 1

Degree course: 
Corso di Second cycle degree in PHYSICS
Academic year when starting the degree: 
2016/2017
Year: 
1
Academic year in which the course will be held: 
2016/2017
Course type: 
Supplementary compulsory subjects
Language: 
Italian
Credits: 
8
Period: 
First Semester
Standard lectures hours: 
80
Detail of lecture’s hours: 
Lesson (80 hours)
Requirements: 

It is useful to have followed (and passed the exams of)
Algebra Lineare e Geometria, Algebra 1, Analisi1, Analisi2.

Final Examination: 
Orale
Assessment: 
Voto Finale

Aims of the course and learning skills

Acquisition of the basilar notions of General Topology; n particular the student has to understand the concepts of connection, compactness, and numerability properties of topological spaces. Ability to recognize in concrete cases the topological properties of a space, and the continuity of maps between spaces. Understanding of some basic concepts of Algebraic Topology: in particular omotopy, retraction, and basic techniques for the computation of the fundamental group of a space.

Contents and course program

General Topology:
Topological spaces and their bases. Remindings of metric spaces. Metrizable topologies.
Hausdorff spaces.
Subspace topology.
Internal part, closure, border of a subspace and their properties.
Continuity between topological spaces.
Product topology.
Separation axioms.
Quoteint topology. Group actions on a topological space. Projective spaces.
Connection and arc connection.
Compactness.
Numerability axioms.
Alexandroff compactification.

Elements of Algebraic Topology:
Omotopy of functions and deformation retraction. Omotopical equivalence of topological spaces.
Fundamental group of a pointed space.
The fundamental group of the circle.
A simplified version of the Theorem of Seifert Van Kampen.
Application: the fundamental group of the spheres.

Teaching methods

Lessons and exercise classes

Textbooks and didactic material

1) M. Manetti, Topologia. Springer, 2008.
2) C. Kosniowski, Introduzione alla topologia algebrica. Zanichelli, 2004 (l'ultima edizione).
Another useful reference is Munkres, Topology (in English).
For exercises and old exams see personal the webpage of the Teacher.

Type of examination

Written and oral exam

Borrowed from

click on the activity card to see more information, such as the teacher and descriptive texts.

Degree course in: MATHEMATICS