Algebraic Topology (FIRST PART)
A.Y. 2024/2025
Learning objectives
(first part) 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
(first part) 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.
Lesson period: First semester
Assessment methods: Esame
Assessment result: voto verbalizzato in trentesimi
Single course
This course can be attended as a single course.
Course syllabus and organization
Single session
Responsible
Lesson period
First semester
Course syllabus
Introduction to homological algebra (chain complexes, exact sequences, Euler characteristic).
Singular homology groups, geometric meaning of H0 and H1.
Pair of spaces: relative homology and its exact sequence.
Connecting homomorphism. Mayer Vietoris exact sequence.
Examples of computation od singula homology groups. Homology of hyperspheres.
Local homology. Invariance of dimension and of the boundary. Generalized Jordan separation theorem.
CW-complex of finite type. The cellular homology complex. Examples of cellular homology.
Cup product. The cohomology ring. Examples.
The universal coefficient theorem.
The de Rham theorem.
Singular homology groups, geometric meaning of H0 and H1.
Pair of spaces: relative homology and its exact sequence.
Connecting homomorphism. Mayer Vietoris exact sequence.
Examples of computation od singula homology groups. Homology of hyperspheres.
Local homology. Invariance of dimension and of the boundary. Generalized Jordan separation theorem.
CW-complex of finite type. The cellular homology complex. Examples of cellular homology.
Cup product. The cohomology ring. Examples.
The universal coefficient theorem.
The de Rham theorem.
Prerequisites for admission
Contents of the courses Geometria 1,2,3,4, and 5
Teaching methods
Traditional: lessons anche class exercizes
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.
- 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.
MAT/03 - GEOMETRY - University credits: 6
Practicals: 12 hours
Lessons: 35 hours
Lessons: 35 hours
Professors:
Bertolini Marina, Turrini Cristina
Shifts:
Professor(s)
Reception:
by appointment (by e-mail)
Math. Dept. - via C. Saldini 50 - Milano