2024/2025
Research Seminar "Localisation and Rational Homotopy Theory"
Category 'Best Course for New Knowledge and Skills'
Type:
Optional course (faculty)
Delivered by:
Faculty of Mathematics
Where:
Faculty of Mathematics
When:
1, 2 module
Open to:
students of all HSE University campuses
Instructors:
Alexei Gorinov
Language:
English
ECTS credits:
6
Course Syllabus
Abstract
The homotopy category of CW-complexes is quite complicated. For example, given two simply-connected polyhedra, it is not clear how to tell whether they are homotopy equivalent, or to describe the homotopy classes of maps between them. It turns out however that these and other similar questions become much easier if we declare maps that induce isomorphisms of rational homology to be isomorphisms. The result is the {\it rational homotopy category}, in which the homotopy classes of maps and the homotopy equivalence classes of objects can be explicitly described.
Course Contents
- Eilenberg – MacLane spaces and Postnikov towers
- Localising a space in a set of primes. Rationalisation.
- Sullivan models of commutative differential graded algebras: existence and uniqueness
- Examples of Sullivan models.
- Applications of Sullivan models: homotopy groups; formality of Kähler manifolds; mapping class groups
- Other models of rational homotopy types (𝐶∞-algebras, Lie models etc.)
- Localisation of a category. Bousfield localisation
