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Lecture# Bar Construction: Homology Groups and Classifying Space

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.

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Instructor

In course

MATH-506: Topology IV.b - 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 a

Related concepts (432)

H

H, or h, is the eighth letter in the Latin alphabet, used in the modern English alphabet, including the alphabets of other western European languages and others worldwide. Its name in English is aitch (pronounced eɪtʃ, plural aitches), or regionally haitch heɪtʃ. The original Semitic letter Heth most likely represented the voiceless pharyngeal fricative (ħ). The form of the letter probably stood for a fence or posts. The Greek Eta 'Η' in archaic Greek alphabets, before coming to represent a long vowel, /ɛː/, still represented a similar sound, the voiceless glottal fricative /h/.

H-dropping

H-dropping or aitch-dropping is the deletion of the voiceless glottal fricative or "H-sound", [h]. The phenomenon is common in many dialects of English, and is also found in certain other languages, either as a purely historical development or as a contemporary difference between dialects. Although common in most regions of England and in some other English-speaking countries, and linguistically speaking a neutral evolution in languages, H-dropping is often stigmatized as a sign of careless or uneducated speech.

Sheaf (mathematics)

In mathematics, a sheaf (: sheaves) is a tool for systematically tracking data (such as sets, abelian groups, rings) attached to the open sets of a topological space and defined locally with regard to them. For example, for each open set, the data could be the ring of continuous functions defined on that open set. Such data is well behaved in that it can be restricted to smaller open sets, and also the data assigned to an open set is equivalent to all collections of compatible data assigned to collections of smaller open sets covering the original open set (intuitively, every piece of data is the sum of its parts).

H with stroke

Ħ (minuscule: ħ) is a letter of the Latin alphabet, derived from H with the addition of a bar. It is used in Maltese for a voiceless pharyngeal fricative consonant (corresponding to the letter heth of Semitic abjads: ح, ח). Lowercase ħ is used in the International Phonetic Alphabet for the same sound. In Unicode, the special character ħ (U+210F), represents the reduced Planck constant of quantum mechanics. In this context, it is pronounced "h-bar". The lowercase resembles the Cyrillic letter Tshe (ћ), or the astronomical symbol of Saturn (♄).

Mathematical proof

A mathematical proof is a deductive argument for a mathematical statement, showing that the stated assumptions logically guarantee the conclusion. The argument may use other previously established statements, such as theorems; but every proof can, in principle, be constructed using only certain basic or original assumptions known as axioms, along with the accepted rules of inference. Proofs are examples of exhaustive deductive reasoning which establish logical certainty, to be distinguished from empirical arguments or non-exhaustive inductive reasoning which establish "reasonable expectation".

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