Scheda programma d'esame
ELEMENTI DI TOPOLOGIA ALGEBRICA
RICCARDO BENEDETTI
Anno accademico2016/17
CdSMATEMATICA
Codice054AA
CFU6
PeriodoPrimo semestre
LinguaItaliano
CdSMATEMATICA
Codice054AA
CFU6
PeriodoPrimo semestre
LinguaItaliano
Moduli  Settore  Tipo  Ore  Docente/i  
ELEMENTI DI TOPOLOGIA ALGEBRICA  MAT/03  LEZIONI  48 

Programma non disponibile nella lingua selezionata
Learning outcomes
Knowledge
The student who successfully completes the course will have the ability to
compute the homology and cohomology of simplicial and cellular complexes;
(s)he will be able to demonstrate a solid knowledge of the applications of
homology theory; (s)he will display advanced knowledge of PoincarĂ©
duality and of the notions of orientation and degree of a map between manifolds.
(s)he will master the algebraic notions of Tor and Ext and know the statements in homology and cohomology of the universal coefficients theorem.
Assessment criteria of knowledge
The student will be asked to compute the homology and the cohomology ring of some topological spaces that can be realized as simplicial or cellular complexes. He will have to demonstrate a solid knowledge of homology and cohomology theory and
their applications.
Methods:
 Final oral exam
Teaching methods
Delivery: face to face
Learning activities:
 attending lectures
 individual study
 Bibliography search
Attendance: Advised
Teaching methods:
 Lectures
Syllabus
Basics of category theory. Simplicial complexes and their homology. First homology as Abelianization of the fundamental group. Dependence on the homotopy type only. Smooth and PL manifolds, with and without boundary. nhomology of an nmanifold. Degree and its applications. Relative homology and the axioms of homology. Homology of Deltacomplexes. Singular homology. Homology with coefficients in a group.
Tor and the universal coefficients theorem for homology. Cohomology. Ext and
the universal coefficients theorem for cohomology. Cup product and PoincarĂ© duality.
Bibliography
Textbooks that will be used extensively:
Hatcher  Algebraic topology
Matveev  Lectures on algebraic topology
Other reference textbooks
Massey  A basic course in algebraic topology
Munkres  Elements of algebraic topology
Spanier  Algebraic topology
Greenberg, Harper  Algebraic topology. A first course
Ultimo aggiornamento 14/11/2016 17:27