Algebraic Topology
A.Y. 2020/2021
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.
Lesson period: First semester
Single course
This course cannot be attended as a single course. Please check our list of single courses to find the ones available for enrolment.
Course syllabus and organization
Single session
Responsible
Lesson period
First semester
The course will be held online with videotaped lessons (synchronously or asynchronously) and uploaded on the Ariel website of the course. Student reception meetings will be organized on the Zoom or Teams platform. All further information will be published on Ariel.
The final examination will be held on platform Teams or Zoom.
The final examination will be held on platform Teams or Zoom.
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.
Final marks are given using the numerical range 0-30, and will be communicated immediately after the oral examination.
Final marks are given using the numerical range 0-30, and will be communicated immediately after the oral examination.
Topologia Algebrica (prima parte)
Course syllabus
Algebraic Topology (first part)
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.
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.
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.
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.
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 and class exercises
Teaching Resources
- J. Milnor, Morse Theory, Annals Study 51. Princeton Univ. Press, Princetone, 1963.
- V. Guillemin, A. Pollack - Differential Topology. AMS Chelsea Publ. 2010.
- V. Guillemin, A. Pollack - Differential Topology. AMS Chelsea Publ. 2010.
Modules or teaching units
Topologia Algebrica (prima parte)
MAT/03 - GEOMETRY - University credits: 6
Practicals: 24 hours
Lessons: 28 hours
Lessons: 28 hours
Professors:
Bertolini Marina, Turrini Cristina
Topologia Algebrica mod/01
MAT/03 - GEOMETRY - University credits: 3
Lessons: 21 hours
Professors:
Bertolini Marina, Turrini Cristina
Professor(s)
Reception:
by appointment (by e-mail)
Math. Dept. - via C. Saldini 50 - Milano