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2017/2018

## Научно-исследовательский семинар "Введение в теорию когомологий"

Статус: Дисциплина общефакультетского пула
Когда читается: 3, 4 модуль
Язык: английский
Кредиты: 5

### Course Syllabus

#### Abstract

One of the main goals of algebraic topology is to answer the question whether two given topological spaces are homeomorphic or homotopy equivalent. This question and several related ones arise not only in topology, but also in mathematical physics, algebra and geometry of any kind. Classical cohomology and generalisations such as K-theory etc. is one of the main computational tools that in some cases allow one to answer this question

#### 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

• Introduction. How to calculate the homology groups of surfaces
• Singular homology. Basic homological algebra: exact sequences, complexes, 5-lemma, homotopy
• Homological algebra continued: acyclic models
• First applications of acyclic models: homotopy invariance and excision for singular cohomology
• CW-complexes, cellular homology and its particular cases and analogues. Simplicial complexes and simplicial homology. Smooth manifolds, Morse functions, handle decompositions and Morse homology
• Homology and cohomology with coefficients. The universal coeffcient theorems
• The Künneth isomorphisms
• Cup and cap products. Topological manifolds and the Poincare duality
• Lefschetz theorems. The contribution of a nondegenerate fixed point in the manifold case
• Vector bundles and characteristic classes
• Basic complex K-theory. Bott periodicity
• An aside: generalised (co)homology. Spectra. Fibration and cofibration sequences
• Applications of the complex K-theory: Hopf invariant 1, vector bundles and complex structures on spheres etc

• Exam

#### Interim Assessment

• Interim assessment (4 module)
0.7 * Exam + 0.3 * Task