34673 - Advanced Geometry 2

Learning outcomes

At the end of the course, the student reaches advanced knowledge in the field of algebraic topology, necessary for the understanding of various mathematical problems. He is able to compute cohomological and homotopic invariants of spaces of ordered and non-ordered configurations, including the group of braids.

Course contents

The aim of this course is to provide tools such as sheaves and spectral sequences for the computation of cohomology groups. They are used in various fields of mathematics: in complex geometry, differential, commutative algebra, algebraic geometry and algebraic topology.

The course is divided into two parts: in the first one we will briefly recall the definitions of homology and cohomology of a topological space and their main properties. We will see some main theorems (e.g. Kunneth, Lefschetz fixed point, Poincaré duality) and their applications to the computation of (co)homology groups.
In the second part we introduce the two important tools: sheaf theory and spectral sequences. Finally, we will apply these tools in algebraic topology by making some examples concerning configuration spaces.

Readings/Bibliography

Hatcher - Algebraic topology
Hatcher - Spectral sequences
Iversen - Cohomology of sheaves

Teaching methods

Frontal lesson with examples and exercises. Some exercises will be left as homework to become familiar with the tools introduced.

Assessment methods

Oral exam or seminar.

Office hours

See the website of Roberto Pagaria