28369 - Geometry 2

Learning outcomes

At the end of the course, students understand the basic notions of general topology and homotopy theory. They are able to use these notions in other fields of mathematics.

Course contents

The course will be totally devoted to Topology.

Category theory: functors, products and coproducts, commutative diagrams.

Metric spaces: completeness and compactness under limits, completion theorem, compactness if and only if the space is complete and totally bounded. Function spaces with metric codomain, Ascoli-Arzelà theorem.

Topological spaces: continuity, connectedness, path connectedness and compactness. Separation axioms. Topology of projective spaces over a field. Induced topology, product and quotient topologies. Separation properties of quotients, connected components and \pi_0 functor. Homotopy theory, fundamental group and coverings: lifts of paths, van Kampen theorem and monodromy of a covering.

Readings/Bibliography

Main book:

Topology by M. Manetti

Auxiliary texts:

Topologia by S. Francaviglia

Geometria 2 by E. Sernesi

Teaching methods

Lectures and exercises in class by the professors, together with the support of a tutor.

Assessment methods

The written exam will be divided in two parts: in the first, students have one hour to answer to three True/False questions, proving the statement in case it is deemed to be true, and producing counterexamples otherwise. Every question gives up to 3 points, and this part is passed with 5 points out of 9. Passing the first part is required for the second part.

The second part consists in the resolutions of problems and exercises and lasts 120 minutes. In this part students can obtain up to 21 points. The final mark of the written exam is obtained by summing the points of the two parts, and admission to the oral exam is obtained with 16 points out of 30.

After passing the written exam, there is an oral colloquium, aimed at assessing the knowledge of the topics and the reasoning abilities. The final mark is determined by the oral colloquium.

In both parts of the written exam, students can have a single protocol sheet (or two A4 ones), handwritten in a reasonable size (no more than one line per square), containing any result that they deem necessary for the written exam. This sheet can be different in the first and second parts. No other material (books, notes) is allowed.

The admission to the oral exam is subordinated to the success in the written part

Teaching tools

There will be support by a tutor

Office hours

See the website of Giovanni Mongardi

See the website of Andrea Petracci