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ADVANCED GEOMETRY 1 (536SM)

A.A. 2019 / 2020

Periodo 
Primo semestre
Crediti 
6
Durata 
48
Tipo attività formativa 
Caratterizzante
Percorso 
[PDS0-2018 - Ord. 2018] comune
Syllabus 
Lingua insegnamento 

English

Obiettivi formativi 

Knowledge and understanding: understanding the most important homology and coomology theories in algebraic topology, knowing the main applications of these theories, acquiring basic notions of homological algebra.

Applying knowledge and understanding: computing homology and coomology groups of topological spaces, solving problems in algebraic topology by using homology and coomology theories.

Learning skills: reading and understanding advanced texts in algebraic topology concerning homology or coomology theories.

Communication skills: presenting in a correct and appropriate manner definitions and theorems included in a homology or a coomology theory.

Prerequisiti 

General topology, basic algebra.

Contenuti 

Algebraic topology: simplicial and singular homology, cellular homology, cohomology

Metodi didattici 

During the lesson theoretical aspects and exercises are presented by using the blackboard. Students are strongly invited to actively take part to the lessons. Regularly I assign to the students some exercises as homework and then solutions are discussed in class.

Programma esteso 

Simplicial homology: simplices and simplicial complexes; simplicial homology groups; zero dimensional homology groups; homology of cones.

Singular homology: basic homological algebra; singular homology groups; topological invariance; zero dimensional homology groups; homology of a point; homotopy invariance; long exact homology sequence; relative homology; excision; Mayer-Vietoris exact sequence; Eilenberg–Steenrod axioms; homology of spheres; fixed point theorem of Brower; degree of maps between spheres; generalized Jordan curve theorem and invariance of domain; isomorphism between singular and simplicial homology; Euler characteristic; homology with coefficients; the Tor functor; the universal coefficient theorem for homology.

Cellular homology: CW complexes; cellular homology groups; equivalence with singular homology; cellular boundary formula; classification and homology of surfaces.

Cohomology: cohomology groups; the Ext funtor; universal coefficient theorem for cohomology; Poincaré duality.

Modalità di verifica dell'apprendimento 

Six possible dates, each consisting of an oral exam. The student has to be able to explain the constructions and the definitions introduced during the course and to present the proof of the theorems which I explained during the lessons

Altre informazioni 

Homework assignments can be found in my webpage: http://www.dmi.units.it/~mecchia/.

Testi di riferimento 

Main textbooks:

J. R. Munkres, Elements of Algebraic Topology. Addison-Wesley Publishing Company 1984

A. Hatcher, Algebraic Topology. Cambridge University Press 2002 (http://www.math.cornell.edu/~hatcher/AT/ATpage.html)


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