Algebraic topology (first part)

A.Y. 2018/2019
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.
Expected learning outcomes
Know how to use some of the algebraic topology techniques on topological spaces and in particular on topological manifolds.
Course syllabus and organization

Single session

Responsible
Lesson period
First semester
Course 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.
MAT/03 - GEOMETRY - University credits: 6
Lessons: 42 hours
Professor(s)
Reception:
by appointment (by e-mail)
Math. Dept. - via C. Saldini 50 - Milano