On the Second Homology of Crossed Modules
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On the Second Homology of Crossed Modules Tahereh Fakhr Taha1 · Hajar Ravanbod1 · Ali Reza Salemkar1 Received: 27 July 2019 / Revised: 23 August 2020 / Accepted: 25 August 2020 © Malaysian Mathematical Sciences Society and Penerbit Universiti Sains Malaysia 2020
Abstract In this article, we present a new description of the integral second homology of crossed modules of groups and generalize two basic results on the integral second homology of groups for crossed modules. Using these, we strengthen some consequences on covering pairs and the universal relative central extensions of pairs of finite groups. Keywords Crossed module · Exterior product · Second homology · Covering pair Mathematics Subject Classification 18G05 · 18G99 · 20J99
1 Introduction Several definitions for the homology of crossed modules have been given during the last years: Ellis [7] and Baues [2] introduced the homology of a crossed module to be the homology of its classifying space. Grandjeán and Ladra [10] defined the second homology crossed module by means of a Hopf formula applied to a particular kind of presentations called ε-projective. Also, associated with an extension of crossed modules, they [16] gave the construction of a five-term exact sequence for the homology of crossed modules. In continuation, Pirashvili [21] presented the notion of the tensor product of two abelian crossed modules, and he used it to construct the Ganea map, that is, extended the above five-term exact sequence one term further. Carrasco et al. [6], using the general theory of cotriple homology of Barr and Beck, defined the integral homology crossed modules of a crossed module as the simplicial derived functors of the abelianization functor from the category of crossed modules to the category of abelian crossed modules and generalized some classical results of the homology of
Communicated by Rosihan M. Ali.
B 1
Ali Reza Salemkar [email protected] Faculty of Mathematical Sciences, Shahid Beheshti University, G.C., Tehran, Iran
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groups. Considering projective presentations introduced in [6] instead of ε-projective presentations, they also obtained the results given in [10,16]. Recently, authors [22] introduced the notions of the non-abelian tensor and exterior products of two normal crossed submodules of some crossed module of groups, which are generalizations of the works of Brown and Loday [4,5] and Pirashvili [21]. In this article, we give a new description of the second homology of crossed modules; in fact, we describe the second homology of crossed modules as the central crossed submodules of their exterior products, which generalizes a result of Miller [18] for groups. Also, we show that the second homology of the direct product of two crossed modules is isomorphic to the direct product of the second homology of the factors and the tensor product of the two crossed modules abelianized. Finally, we give some applications to covering pairs and the universal central extensions of pairs of finite groups.
2 Preliminaries on Crossed
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