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Обычная версия сайта
2018/2019

## Научно-исследовательский семинар "Гомотопическая топология"

Лучший по критерию «Полезность курса для расширения кругозора и разностороннего развития»
Статус: Дисциплина общефакультетского пула
Когда читается: 1, 2 модуль
Язык: английский

### Программа дисциплины

#### Аннотация

We give an introduction to generalised cohomology and stable homotopy theory. At first, we consider examples and a few applications of generalised homology and cohomology, such as the Bott periodicity, Hopf invariant 1, complex structures on spheres, representing classes by manifolds, cobordism rings. After that we develop a general theory: spectra, stable homotopy category, fibration and cofibration sequences, the Whitehead theorem, the Atiyah duality.

#### Цель освоения дисциплины

• The seminar is intended to introduce the subject area to the students, and to offer them an opportunity to prepare and give a talk.

#### Результаты освоения дисциплины

• Successful participants improve their presentation skills and prepare for participation in research projects in the subject area.

#### Содержание учебной дисциплины

• Axioms for generalized (co)homology.
• Cofibration sequences for spaces. Omega-spectra and cohomology theories
• Fibration sequences for spaces
• First applications: the Dold–Thom theorem, representing rational homotopy classes by manifolds.
• Brown’s representability theorem for cohomology.
• Basic K-theory.
• Complex Bott periodicity; extending the complex K-theory to a cohomology theory.
• Applications of K-theory: the Hopf invariant 1 and almost complex structures on spheres.
• Spectra and stable homotopy category. Homotopy groups of spectra.
• Thom spectra and cobordism. The Pontrjagin–Thom theorem.
• Calculation of π∗(MO) and π∗ (MSO) ⊗ ℚ.
• Spectra can be desuspended.
• Fibration and cofibration sequences for spectra.
• Duality for spectra. The Alexander duality.
• The Thom isomorphism for generalized cohomology and the Atiyah duality.
• The topological Riemann – Roch theorem and applications. Schwarzenberger’s conditions on the Chern numbers of complex vector bundles on ℂℙ^n.

• Final exam

#### Промежуточная аттестация

• Промежуточная аттестация (2 модуль)
0.3 * Cumulative grade + 0.7 * Final exam