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Department of Mathematics and Applications
"Renato Caccioppoli"


Geometry 3

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Geometry 3

Credits

6

Prerequisites

Geometry 2.

Scientific-disciplinary sector (SSD)

MAT/03 Geometry.

Examination method

Passing an oral exam.

Learning
objectives

The aims of the course are:

  • to develop and examine critically, in greater depth, some topics in general topology;
  • to acquire the fundamental preliminary tools for the study of topological manifolds;
  • to discuss the main proof techniques in the areas described in the previous points;
  • to acquire the ability to place the notions learnt and the most important results in a more applied context.

Syllabus

Review of general topology. Connected spaces. Compact spaces. Embeddings and subspaces. Topological groups. Exhaustions by compact sets. Identifications and quotient topology. Quotients by groups of homomorphisms. Topological manifolds. Locally connected spaces. The functor p0. Homotopy. Retractions and deformations. Homotopy of paths. The fundamental group. The functor p1. Simple connectedness of the sphere. Local homeomorphisms. Coverings. Quotients by properly discontinuous actions. Sections. Homotopy lifting. The Brouwer and Borsuk theorem. An example of a non-abelian fundamental group. Monodromy of a covering. Group actions on sets. An isomorphism theorem. Lifting of arbitrary maps. Regular coverings. Universal coverings.

Expected learning
outcomes

at the end of the course, students must demonstrate that they:

  • know and understand the topics covered in lectures, are familiar with the language of general topology and are able to illustrate the main proof techniques discussed;
  • are able to apply the knowledge acquired in the study and solution of problems of varying complexity;
  • are able to communicate ideas and solutions clearly, rigorously and effectively to both specialist and non-specialist audiences;
  • are able to identify the most appropriate methods to analyse and solve a problem related to the course topics and to interpret the results correctly.

Learning outcomes
to be assessed

Assessment will cover the knowledge and skills acquired on the topics developed during the course, mastery of the mathematical tools used, presentation skills and appropriate use of language, the ability to apply the knowledge acquired to the solution of simple problems, the ability to place that knowledge in more applied areas, and the ability to support a discussion with examples and counterexamples.