Algebraic topology (first part)

A.Y. 2018/2019
Lesson for
6
Max ECTS
42
Overall hours
SSD
MAT/03
Language
Italian
Learning objectives
The aim of the course is to introduce the main results and to provide some of the techniques of algebraic topology, with some hints to differential topology.
Know how to use some of the algebraic topology techniques on topological spaces and in particular on topological manifolds.

Course structure and Syllabus

Active edition
Yes
Responsible
MAT/03 - GEOMETRY - University credits: 6
Lessons: 42 hours
Syllabus
Singular homology. Geometric meaning of H0 e H1.
Topological pairs and relative homology. The long exact sequence in relative homology. Mayer Vietoris exact sequence.
Examples. Applications of the homology of spheres.
The invariance of dimension and of the boundary. Generalized Jordan curve theorem.
CW-complex of finite type. The cellular homology complex. Examples of cellular homology.
Morse Functions. Morse Lemma. Morse theorems I and II. Reeb theorem. Differentiable compact manifolds and CW complexes . Morse's inequality and equality.
Cup product. The cohomology ring. Examples. The universal coefficient theorem.
Lesson period
First semester
Lesson period
First semester
Assessment methods
Esame
Assessment result
voto verbalizzato in trentesimi
Professor(s)
Reception:
by appointment (by e-mail)
Math. Dept. - via C. Saldini 50 - Milano