Select Academic Year:     2016/2017 2017/2018 2018/2019 2019/2020 2020/2021 2021/2022
Professor
GIANLUCA BANDE (Tit.)
Period
Second Semester
Teaching style
Convenzionale
Lingua Insegnamento
ITALIANO

Informazioni aggiuntive

Course Curriculum CFU Length(h)
[60/65]  MATHEMATICS [65/40 - Ord. 2020]  MATEMATICA PURA 9 72

### Objectives

KNOWLEDGE AND SKILLS 'UNDERSTANDING: learning the basic concepts of algebraic topology such as homothopy, cethegory anf functor, fundamental group and singular homology of a topological space.

CAPACITY OF APPLICATION: the student must be able to apply all the general knowledge necessary for the understanding of algebraic topology to recognize and analyzing the main topological spaces.

JUDGEMENT: The course aims to stimulate objective teaching evaluation constantly offering students a comparison between the theoretical contents proposed while they were lessons front and obtain them through self-study using the recommended texts and course material provided.

COMMUNICATION SKILLS: ability to express with the appropriate mathematical terminology the basic concepts of algebraic topology.

LEARNING SKILLS:
Development of professional communication skills in the mathematical field through the use of correct terminology and an organized and comprehensible description method useful not only for passing the exam but also for a preparatory preparation for subsequent examinations ( In particular of geometry and algebra). Ability to find topological properties of not too complicated spaces and to use this knowledge to distinguish two topological spaces. Ability to read and understand basic texts of algebraic topology.

### Prerequisites

Knowledge of the the first degree in Mathematics

### Contents

Homothopy of continuous functions; mechanical interpretation of homothopy thoery; examples of homothopic and non-homothopic functions; antipodel functions from the sphere to itself; homothopy beteween arcs; relative homothopy; the space of continuous functions and the compact-open topology; stability of homothopy with respect to continuous functions; a continuous function into the circle is homothopic to the constant function if and only if it extends to a continuous function to the disk; spaces homothopically equivalent; convex spaces; contractible spaces; the circle is not contractible; deformation retracts; examples; the figure eight; the fugure theta; a Theorem od Fuchs only statement (two topological spaces are homothopically equivalenti if and only if they are homeomorphic to deformation retracts of the same topological space); products of arcs and its main properties; the fundamental group; dependence of the fundamental group from its base point; induced homeomorphism form a continuous functions between topological spaces; homothopic invariance of the fundamental group; simply-connected spaces; fundamnetal group of the product of twio topological spaces; computation of the fundamental group of the circle; fundamental theorem of algebra; Brower fixed point theorem for the bidimensional disk; a theorem of Frobenius (a 3X3 matrix with positive real entries admits a positive eigenvalue); Peano’s curves; Lebesgue number; the n-dimensional sphere is simply-connected for n>1; free groups; presentation of a group; examples; Seifert Van Kampen theorem (proof for generators and statement for relations); a psace which is the union of two simply-connected open sets with non-empty simply-connected intersection is simply-connected; computation of the fundamental group of the complement of a finite number of points in the n-dimensional Euclidean space; the fundamnetal group of teal and complex projecitve spaces; the fundamental group of a manifold of dimension greater or equal than twois is isomorphic to the the fundamental group of the manifoldminus one of its point; the the fundamental group
Of the complement of a line or a circle in the space; the the fundamental group of n circles woth a point in common; the the fundamental group of the eight figure; classification of surfaces (with or without boundary) and computation of thier fundamental group; definition of covering; tota space e fiber of point; examples and non examples of covering spaces; coverings and local homeomorphisms; a local homeomorphism between a compact Haussdorf space and a connected space is a covering; lift of a continuous function; unicity of the lift of a continuous function with connected domain; lfting arc; if the base of a covering is connected the the cardinality of the fiber does not depend on hte point; lift of homothopy; if the total space of a covering is connected by arcs and the base is simply-connected then the covering is a homeomorphism; actions of groups on topological spaces; properly discontinuous functions and covering spaces; examples; lens spaces; covering spaces and fundamental groups; monodromy of a covering space; the fundamental group of an orbit space; theorems on lifting properties for covering spaces; universal covering spaces. Simplexes. Singular homology. First properties: connected components, arcwise connected components, point homology. Chain complexes. Homotopy invariance. Fundamental group and 1st homology group. Exact sequences, relative homology and excision. Homology of the sphere and Brower fixed point. Invariance of dimension. Degree, vector fields on spheres. Mayer-Vietoris sequence. Homomorphism of Hurewicz.

### Teaching Methods

The teaching will be provided mainly in the classical way and, as foreseen in the Manifesto of Studies for the A. 2021-22, integrated and “augmented” with online strategies, in order to guarantee their use in an innovative and inclusive way.

Blackboard and slides during the lectures. Exercises with the students during the lectures.

### Verification of learning

Compatibly with the indications of the University on how to carry out the exams according to the evolution of the COVID-19 emergency, the exams could be held in the presence or online.

The oral exam consists of a discussion of at least three arguments. The student must demonstrate that they have understood and assimilated the arguments (knowledge and understanding, knowledge and understanding applied). In addition, students must demonstrate that they are able to explain the concepts and demonstrations learned during the course (Communication Skills). To this end it is necessary that the student, during the oral exam, write on the chalkboard and explain all the steps that it is following to reach the conclusion of an argument (independent judgment). It is advisable to answer exactly the questions without rambling and writing immediately all the necessary details (independent judgment, communication skills)
In any case an answer showing strong lacks on the knowledge of course contents or basic notions could compromise the entire oral test.

### Texts

Textbooks
-Notes of Bande-Loi (textbook for fundamental group and homotopy).
- J.W. Vick, "Homology Theory. An introduction to Algebraic topology" (Singular homology)
Other books

-M.J. Greenberg, J.R. Harper, “Algebraic topology. A first course” Mathematics Lecture Note Series, 58. Benjamin/Cummings Publishing Co., Inc., Advanced Book Program, Reading, Mass., 1981. (Singular homology )
-C. Kosnioski, “Introduzione alla Topologia Generale-Zanichelli” (Singular homology )
-A. Hatcher, “Algebraic Topology” (Singular homology)
-Y. Félix, D. Tanré, "Topologie Algébrique - Cours et exercices corrigés"