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Regular version of the site
2020/2021

# Research Seminar "Homothopy Theory"

Category 'Best Course for Career Development'
Category 'Best Course for Broadening Horizons and Diversity of Knowledge and Skills'
Category 'Best Course for New Knowledge and Skills'
Type: Optional course (faculty)
When: 1, 2 module
Instructors: Alexei Gorinov
Language: English
ECTS credits: 6
Contact hours: 60

### Course Syllabus

#### Abstract

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.

#### Learning Objectives

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

#### Expected Learning Outcomes

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

#### Course Contents

• 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

#### Interim Assessment

• Interim assessment (2 module)
0.3 * Cumulative grade + 0.7 * Final exam