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.
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.
Topologia Algebrica (prima parte)
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.
Traditional: lessons anche class exercizes
- 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
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.
Traditional: lessons and class exercises
- J. Milnor, Morse Theory, Annals Study 51. Princeton Univ. Press, Princetone, 1963. - V. Guillemin, A. Pollack - Differential Topology. AMS Chelsea Publ. 2010.