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Irish Mathematical Society Bulletin
Article . 1988 . Peer-reviewed
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Crossed Modules

Crossed modules
Authors: Ellis, Graham;

Crossed Modules

Abstract

Crossed modules provide a very useful tool in algebraic topology, algebraic K-theory, group cohomology and combinatorial group theory. They were introduced almost 40 years ago by J. H. C. Whitehead in his work on combinatorial homotopy theory. Given this illustrious start and their proven potential, it is strange that they are not very well known to the relevant sections of the mathematical fraternity. This article aims to introduce the general mathematical audience to this very elegant group theoretic structure. The article is well written, thorough and although assuming little or no acquaintance with algebraic topology, etc., it manages to get over the main points of the theory and use of crossed modules. It can be thoroughly recommended to anyone wishing to see what crossed modules are, why they are useful and how they arise. It is to be hoped that the limited circulation of the journal concerned will not inhibit the distribution of this article - perhaps an enlarged version might be published elsewhere.

Keywords

Homological methods in group theory, Nonabelian homological algebra (category-theoretic aspects), group cohomology, Group rings, combinatorial group theory, Research exposition (monographs, survey articles) pertaining to algebraic topology, Group rings of infinite groups and their modules (group-theoretic aspects), crossed modules, homotopy theory, survey, Research exposition (monographs, survey articles) pertaining to group theory, Groupoids (i.e. small categories in which all morphisms are isomorphisms)

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    popularity
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    influence
    This indicator reflects the overall/total impact of an article in the research community at large, based on the underlying citation network (diachronically).
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    impulse
    This indicator reflects the initial momentum of an article directly after its publication, based on the underlying citation network.
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selected citations
These citations are derived from selected sources.
This is an alternative to the "Influence" indicator, which also reflects the overall/total impact of an article in the research community at large, based on the underlying citation network (diachronically).
BIP!Citations provided by BIP!
popularity
This indicator reflects the "current" impact/attention (the "hype") of an article in the research community at large, based on the underlying citation network.
BIP!Popularity provided by BIP!
influence
This indicator reflects the overall/total impact of an article in the research community at large, based on the underlying citation network (diachronically).
BIP!Influence provided by BIP!
impulse
This indicator reflects the initial momentum of an article directly after its publication, based on the underlying citation network.
BIP!Impulse provided by BIP!
2
Average
Average
Average
gold