Algebraic topology

A.Y. 2020/2021
Overall hours
Learning objectives
The aim of the course is to introduce the main results and to provide some of the techniques of algebraic topology and of differential topology.
Expected learning outcomes
Know how to use some of the algebraic topology techniques on topological spaces and in particular on topological manifolds, and how to use some of the differential topology techniques on smooth manifolds.
Course syllabus and organization

Single session

Lesson period
First semester
Course syllabus
Singular homology. Geometric meaning of H0 e H1. Topological Euler characteristic.
Topological pairs and relative homology. The long exact sequence in relative homology. The connecting homomorphims. Mayer Vietoris exact sequence. Examples. Applications of the homology of spheres. The invariance of dimension and of the boundary. Generalized Jordan curve theorem.
Topological degree for maps between spheres. CW-complex of finite type. The cellular homology complex. Examples of cellular homology.
Cup product. The cohomology ring. Examples.
The universal coefficient theorem.
Prerequisites for admission
Contents of the courses Geometria 1,2,3,4, and 5.
Teaching methods
Traditional: lessons and class exercises.
Teaching Resources
- M. J. Greenberg, J. R. Harper, Algebraic Topology. A First Course, The Benjamin/Cummings Publishing Company, 1981.
- A. Hatcher, Algebraic Topology, online version.
Assessment methods and Criteria
The final examination consists of an oral exam. In the oral exam, the student will be required to illustrate results presented during the course in order to evaluate her/his knowledge and comprehension of the arguments covered as well as the capacity to apply them. Moreover the student could be asked to solve some exercises.
Final marks are given using the numerical range 0-30, and will be communicated immediately after the oral examination.
MAT/03 - GEOMETRY - University credits: 6
Lessons: 42 hours
by appointment (by e-mail)
Math. Dept. - via C. Saldini 50 - Milano