Topology and Homotopy
Code
10842
Academic unit
Faculdade de Ciências e Tecnologia
Department
Departamento de Matemática
Credits
6.0
Teacher in charge
João Nuno Gonçalves Faria Martins
Total hours
56
Teaching language
Português
Objectives
The aim of this course is to provide the students a course on point set topology, illustrated with numerous explicit examples, also being an introduction to Algebraic Topology and to Homotopy Theory.
The students should understand and know how to prove the fundamental results about topological spaces, continuous functions, compactness, connectedness and separation / countability axioms.
In the field of algebraic topology, the students will learn the construction of the fundamental group of a topological space, and its calculation in simple cases, resulting from particular instances of the van Kampen theorem, and from the use of covering spaces and of homotopy equivalences.
In order to emphasize the importance of functoriality in algebraic topology, a general background on category theory will be provided, including the central notions of products and co-products.
Another objective of this course is to show immediate applications of algebraic topology, for example the Fundamental Theorem of Algebra, the Brouwer Fixed Point Theorem (for the disk) and, time permitting, to give a proof of the Jordan curve theorem.
Prerequisites
Analysis in R^n. Linear algebra. Metric spaces. (Basic notions).
General algebra, equivalence relations, groups, rings, homomorphisms, isomorphisms.
Subject matter
Topological spaces. Basis and sub-basis. Continuous functions. Homeomorphisms. Topological properties.
Connectedness. Arc connectedness. (Arc) Components. Local (arc) connectedness.
Compactness. Sub-basis Theorem. Products of topological spaces. Tychonoff''''s theorem. Compactness in metric spaces. Lebesgue numbers. Local compactness.
Axioms of numerabilidade. Separation axioms. Urysohn''''s Lemma. Tietze extension theorem.
Quotient topology. The projective plane.
Homotopy between curves. Fundamental group. Independence of the base point. Fundamental group of the sphere. Categories and functors. Functoriality of the fundamental group. Brouwer fixed point theorem.
Coverings. Lifting of curves and of homotopies. The fundamental group of the circle and of the projective plane.
The category of topological spaces and of functions up to homotopy. Homotopy equivalence. Applications including a proof of the fundamental theorem of algebra.
Possible additional topics: Jordan curve theorem. Coproducts of groups. Van Kampen theorem.
Bibliography
1) James R. Munkres: Topology. Prentice Hall (2000)
2) Armstrong, Mark Anthony: Basic topology. Corrected reprint of the 1979 original. Undergraduate Texts in Mathematics. Springer-Verlag, New York-Berlin, 1983.
3) Gamelin, Theodore W.; Greene, Robert Everist: Introduction to topology. Second edition. Dover Publications, Inc., Mineola, NY, 1999.
Teaching method
The course will have (per week) 3 hours of theoretical classes and 1 hour of problem classes. The students will be offered 1.5 hours of office hours per week.
Evaluation method
Continuous Assessment
-The students must submit fortnightly series of exercises proposed by the course professor. The weighted average of all the grades will count towards the final mark of the course, with the weight of 75%. Of these grades, it will play an important part the discussion of some of the exercises. This discussion will take place during the problem classes.
-Each student will make an oral presentation of about 1 hour (for example with a complete proof of a theorem, explanation of a small section of a book, etc.) The grade of the presentation will count towards the final mark of the course with the weight of 25%.
Final Exam
Frequency: A student obtains frequency if he has submitted all the series of exercises proposed, except possibly for one.