Algebraic Topology

A.Y. 2021/2022
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
More specific information on the delivery modes of training activities for academic year 2021/22 will be provided over the coming months, based on the evolution of the public health situation.
Prerequisites for admission
Contents of the courses Geometria 1,2,3,4, and 5
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.
Topologia Algebrica (prima parte)
Course syllabus
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.
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.
Topologia Algebrica mod/01
Course syllabus
Differential Topology
Morse Functions. Morse Lemma. Morse theorems I and II. Reeb theorem. Differentiable compact manifolds and CW complexes . Morse's inequality and equality. Examples.
Intersection theory for differential varieties and oriented varieties. Intersection numbers and homotopy invariance property.
Brower degree for maps between spheres. Selfintersection and Eulero Poincaré characteristic of a smooth orientable manifold. Isolated zeroes of vector fields and their indexes. Poincaré Hopf theorem.
Teaching methods
Traditional: lessons anche class exercizes
Teaching Resources
- J. Milnor, Morse Theory, Annals Study 51. Princeton Univ. Press, Princetone, 1963.
- V. Guillemin, A. Pollack - Differential Topology. AMS Chelsea Publ. 2010.
Topologia Algebrica (prima parte)
MAT/03 - GEOMETRY - University credits: 6
Practicals: 24 hours
Lessons: 28 hours
Topologia Algebrica mod/01
MAT/03 - GEOMETRY - University credits: 3
Lessons: 21 hours
Educational website(s)
by appointment (by e-mail)
Math. Dept. - via C. Saldini 50 - Milano