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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

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

Результаты освоения дисциплины

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

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.
  • Whitehead’s theorem for spectra.
  • 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.
Элементы контроля

Элементы контроля

  • неблокирующий Created with Sketch. Cumulative grade
    cumulative grade is proportional to number of tasks solved
  • неблокирующий Created with Sketch. Final exam
Промежуточная аттестация

Промежуточная аттестация

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

Список литературы

Рекомендуемая основная литература

  • Allen Hatcher. (2002). Algebraic topology. Retrieved from http://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=edsbas&AN=edsbas.87FE219C

Рекомендуемая дополнительная литература

  • Atiyah, M. F., & Anderson, D. W. (2018). K-theory. Boca Raton: CRC Press. Retrieved from http://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=edsebk&AN=1728843