Lecture

Bar Construction: Homology Groups and Classifying Space

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In course

MATH-506: Topology IV.a - Cohomology rings

Singular cohomology is defined by dualizing the singular chain complex for spaces. We will study its basic properties, see how it acquires a multiplicative structure and becomes a graded commutative algebra. We study an algebraic version, namely group coho ...

Description

This lecture introduces the bar construction method to compute the first and second homology groups of a group G, construct the classifying space BG, and nearly prove the Hopf formula for the second homology group. The bar resolution is used to construct a free ZG-resolution of the trivial ZG-module. The lecture progresses through the construction of the classifying space BG and discusses the relationship between the homology groups and the classifying space. The presentation concludes with a detailed explanation of the action of G on the cells of the classifying space and the universal cover of BG.

Official source

Ontological neighbourhood

Mathematics

Algebra: Abstract algebra, Representation theory

Topology: Algebraic topology

Health sciences

Immunology: Immune system

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