Homology, Homotopy and Applications

The low-dimensional homology of crossed modules

N. D. Gilbert

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This paper is prompted by work of Grandjeán and Ladra on homology crossed modules. We define the homology of a crossed module in dimensions one and two via the equivalence of categories with group objects in groupoids. We show that for any perfect crossed module, its second homology crossed module occurs as the kernel of its universal central extension as defined by Norrie. Our first homology is the same as that defined by Grandjeán and Ladra, but the second homology crossed modules are in general different. However, they coincide for aspherical perfect crossed modules. Our methods can in principle be applied to define a homology crossed module in any dimension.

Article information

Homology Homotopy Appl., Volume 2, Number 1 (2000), 41-50.

First available in Project Euclid: 13 February 2006

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 20J05: Homological methods in group theory
Secondary: 18D35: Structured objects in a category (group objects, etc.) 18G50: Nonabelian homological algebra 55P15: Classification of homotopy type


Gilbert, N. D. The low-dimensional homology of crossed modules. Homology Homotopy Appl. 2 (2000), no. 1, 41--50. https://projecteuclid.org/euclid.hha/1139841212

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