The student will become familiar with the main themes of homotopy and homology theory. On the homotopy side, they will acquire the ability to describe retractions and homotopic equivalences between spaces, to define covering spaces on them and to link covering spaces with their characteristic subgroup. They also will be able to recognize compact complex surfaces. On the homology side, they will learn the bases of the theory of singular homology, understanding how it satisfies the general homology theory axioms, and they will be capable of describing a topological space as a cell complex, thus computing its main invariants, also using exact sequences and other tools from homological algebra.
Course Prerequisites
The basic notions of group theory, linear algebra and general topology and knowledge of the fundamental group and Van Kampen Theorem.
Teaching Methods
Lectures and problem sessions
Assessment Methods
The exam has - a written part, where items will be graduated so as to identify the competence level of the student with proposed exercises in which the student is asked to describe retractions, homotopic equivalences, covering spaces and cellular complexes, and to compute characteristic subgroups and homology groups, showing that they are able to apply results and concepts of the course; - an oral part, where questions of varying difficulty are asked to the student, who will have to show to have become familiar with the ideas taught in the course and to be able to state and explain them clearly, and to apply them to solve problems proposed on the spot. The evaluation takes into consideration in both parts the shown depth of understanding, clarity and ability to apply tools to new situations. The score will not be an arithmetical mean, or any other automatic formula given the evaluation of the two parts, but will be determined by a global evaluation of the shown preparation.
Texts
In order of relevance: - E. Munkres, Topology, Pearson, - E. Munkres, Elements Of Algebraic Topology, CRC Press - C. Kosniowski, Introduzione alla topologia algebrica. Zanichelli, 2004. - M. Manetti, Topologia. Springer, 2014 (second edition). - A. Hatcher: "Algebraic Topology", Cambridge University Press (freely available online) - M. Greenberg, J. Harper: "Algebraic Topology". - W. Massey: "A Basic Course in Algebraic Topology", Springer-Verlag.
Contents
Jordan curve theorem. Classification of surfaces. Triangulations, Euler-Poincarè characteristic, orientation. Homotopic invariants, retractions and homotopic equivalences. Covering spaces. Basic notions of homological algebra. Singular homology and its homotopic properties, relative homology, axiomatic homology theory. Simplicial homology. Simplicial complexes, CW-complexes. Exact sequences and excision. Equivalence between simplicial and singular homology. Mayer-Vietoris sequence.
Course Language
Italian
More information
Office hours by appointment (send an email) The students belonging to the categories of the project on innovative teaching will have the possibility to attend office hours also in the late afternoon and to see the notes of the lectures.
