GEOMETRY 1
- Overview
- Assessment methods
- Learning objectives
- Contents
- Bibliography
- Teaching methods
- Contacts/Info
It is useful to have attended (and passed the exams of)
Algebra Lineare e Geometria, Algebra 1, Analisi 1, Analisi 2.
Written and oral examination.
The written examination lasts 2 hours and 30 minutes and consists of a number of exercises.
The written examination is graded on a scale from 0 to 30. Students need a grade of 14/30 or more to be admitted to the oral examination.
The oral examination usually begins with the discussion of the written test. The student will then be required to present some of the results seen in class. It will be object of evaluation the ability to present a proof in a complete and rigorous way and to apply these results in concrete cases.
Acquisition of the basilar notions of General Topology; n particular the student has to understand the concepts of connection, compactness, and numerability properties of topological spaces. Ability to recognize in concrete cases the topological properties of a space, and the continuity of maps between spaces. Understanding of some basic concepts of Algebraic Topology: in particular, homotopy, retraction, and basic techniques for the computation of the fundamental group of a space.
Topological spaces and continuous functions. Bases for a topological space. Subspace topology. Interior, closure, border of a subspace and their properties.
Countability and separation axioms. Metric spaces and metrizable topologies.
Product topology.
Quotients. Group actions on a topological space. Projective spaces.
Connectedness and path connectedness.
Compactness. Compactifications of topological spaces.
Elements of Algebraic Topology:
Homotopy of functions and deformation retracts. Homotopical equivalence of topological spaces.
Fundamental group of a pointed space.
The fundamental group of the circle.
A simplified version of the Theorem of Seifert Van Kampen.
Application: the fundamental group of the spheres.
Textbooks and teaching material:
1) M. Manetti, Topologia. Springer, 2008.
2) C. Kosniowski, Introduzione alla topologia algebrica. Zanichelli, 2004 (l'ultima edizione).
Other useful references are:
- Sernesi, Geometria 2
- Munkres, Topology (in English).
- Steen and Seebach, Counterexamples in Topology, Springer-Verlag.
Theoretical lectures and exercise sessions.
To fix an an appointment please send an e-mail to the Lecturer.