OVERVIEW

The course is an introduction to algebraic techniques in topology. In particular, we will introduce singular homology and cohomology, higher homopoty groups and, it time permts, persistent homology. 

AIMS AND CONTENT

LEARNING OUTCOMES

The aim of the course is to provide the student with an elementary introduction to the concepts and methods of Algebraic Topology.

AIMS AND LEARNING OUTCOMES

The aim of this course is to provide students with a consolidation of the techniques of algebraic topology they have already learned in the course Geometry 2, in particular through the theory of universal covering. In addition, the course aims to introducing new concepts such as the theory of CW-complexes and the theory of homology and cohomology,  by using, in particulr, singular homology, Čech cohomology and de Rham cohomology  as fundamental examples, All this without neglecting the motivations and historical background on the emergence of these objects. At the end of the course, the student will be able to calculate coverings of a topological space, quotients of a topological manifolds under the action of a discrete group, or establish whether a given topological space is a CW-complex. Finally, the student will be able to calculate the homology and cohomology groups of simple topological manifolds and calculate their cohomology ring.

PREREQUISITES

The course is a natural continuation of the teaching of Geometry 2. It is advisable to have taken at least one course in: linear algebra and analytic geometry, general algebra, differential geometry.

TEACHING METHODS

The goal of the lectures is to present the theoretical notions and to prove some fundamental theorems, as well as providing solutions to problems, whose aim is to help explain better the theory. Attendance at lectures and exercises is strongly recommended.
Working students and students with certified specific learning disorders (SLD), disabilities, or other special educational needs are encouraged to contact the instructor at the beginning of the course to agree on teaching and assessment methods that, while respecting the learning objectives, take into account individual learning styles

SYLLABUS/CONTENT

1.  Topological manifolds and differentiable manifolds.
2.  Actions of properly discontinuous groups.
3.  Covering spaces 
4. CW - Complexes and their properties.
5. Singular homology and cohomology (in particular, the Mayer-Vietoris theorem and the "universal coefficient theorem", and the Eilenberg-Steenrod  axioms  will be presented).
6. Čech cohomologye and de RhamCohomology
7. Basics of Morse theory
8. Baics of higher homotopy groups

RECOMMENDED READING/BIBLIOGRAPHY

1. C. Godbillon, Topologie algébrique, Hermann, Paris 1971

2. W. S. Messey, A Basic Course in Algebraic Topology , Springer-Verlag, New York 1991.

3. A. Hatcher, Algebraic Topology, Cambridge University Press 2002 (available online).

4. C. A. Weibel, An Introduction to Homological algebra, Cambridge University Press, Cambridge 1994. 

TEACHERS AND EXAM BOARD

LESSONS

LESSONS START

https://corsi.unige.it/corsi/11897

Class schedule

The timetable for this course is available here: Portale EasyAcademy

EXAMS

EXAM DESCRIPTION

Oral Exam.

Students with a certified DSA, disability or other special educational needs are advised to contact the lecturer at the beginning of the course in order to agree on teaching and examination methods that, while respecting the teaching objectives, take into account individual learning methods and provide suitable compensatory tools.

ASSESSMENT METHODS

During the oral examination, the student must be able to: prove the theorems presented in the lecture, correctly state all definitions and solve simple exercises consisting in the calculation of coverings, homology and cohomology.

FURTHER INFORMATION

For any additional information not included in the course description, please contact the instructor