# Homological models for semidirect products of finitely generated Abelian groups

@article{lvarez2012HomologicalMF, title={Homological models for semidirect products of finitely generated Abelian groups}, author={V. {\'A}lvarez and J. Armario and M. Frau and P. Jurado}, journal={Applicable Algebra in Engineering, Communication and Computing}, year={2012}, volume={23}, pages={101-127} }

Let G be a semidirect product of finitely generated Abelian groups. We provide a method for constructing an explicit contraction (special homotopy equivalence) from the reduced bar construction of the group ring of G, $${\overline{B}(\mathsf{\textstyle Z\kern-0.4em Z}[G])}$$ , to a much smaller DGA-module hG. Such a contraction is called a homological model for G and is used as the input datum in the methods described in Álvarez et al. (J Symb Comput 44:558–570, 2009; 2012) for calculating a… Expand

#### Topics from this paper

#### 2 Citations

Formal Orthogonal Pairs via Monomial Representations and Cohomology

- Mathematics
- 2020

A Formal Orthogonal Pair is a pair $(A,B)$ of symbolic rectangular matrices such that $AB^T=0$. It can be applied for the construction of Hadamard and Weighing matrices. In this paper we introduce a… Expand

On higher dimensional cocyclic Hadamard matrices

- Mathematics, Computer Science
- Applicable Algebra in Engineering, Communication and Computing
- 2014

This method provides an uniform way of looking for higher dimensional n-cocyclic Hadamard matrices for the first time. Expand

#### References

SHOWING 1-10 OF 43 REFERENCES

Homology of iterated semidirect products of free groups

- Mathematics
- 1995

Abstract Let G be a group which admits the structure of an iterated semidirect product of finitely generated free groups. We construct a finite, free resolution of the integers over the group ring of… Expand

A Mathematica Notebook for Computing the Homology of Iterated Products of Groups

- Mathematics, Computer Science
- ICMS
- 2006

Computational results provided by the program have allowed the simplification of some of the formulae involved in the calculation of Hn(G), and the efficiency of the method has been improved. Expand

Cohomology of metacyclic groups

- Mathematics
- 1991

group N by a finite cyclic group K . Using homological perturbation theory, we introduce the beginning of a free resolution of the integers Z over the group ring ZG of G in such a way that the… Expand

Computing Resolutions Over Finite p-Groups

- Mathematics
- 2001

A uniform and constructive approach for the computation of resolutions and for (co)homology computations for any finite p-group is detailed. The resolutions we construct ([32]) are, as vector spaces,… Expand

Computing the homology of groups: The geometric way

- Mathematics, Computer Science
- J. Symb. Comput.
- 2012

This work has developed some algorithms which, making use of the effective homology method, construct the homology groups of Eilenberg-MacLane spaces K(G,1) for different groups G, allowing one in particular to determine the homological groups of G. Expand

Calculation of cocyclic matrices

- Mathematics
- 1996

Abstract In this paper we provide a method of explicitly determining, for a given finite group G and finitely generated G -module U trivial under the action of G , a representative for each element… Expand

Perturbation Theory in Differential Homological Algebra II

- 1989

Perturbation theory is a particularly useful way to obtain relatively small differential complexes representing a given chain homotopy type. An important part of the theory is “the basic perturbation… Expand

The twisted Eilenberg-Zilber Theorem

- Mathematics
- 2009

The purpose of this paper is to give a simpler proof of a theorem of E.H. Brown [Bro59], that if F → E → B is a fibre space, then there is a differential on the graded group X = C(B) ⊗Λ C(F ) such… Expand

Cocyclic Development of Designs

- Mathematics
- 1993

AbstractWe present the basic theory of cocyclic development of designs, in which group development over a finite group G is modified by the action of a cocycle defined on G × G. Negacyclic and… Expand

Perturbation Theory in Differential Homological Algebra I

1.1. Introduction. Differential homological algebra extends the classical machinery of homological algebra to differential algebras and modules. As first introduced by Eilenberg and Moore [5], the… Expand