Homological models for semidirect products of finitely generated Abelian groups

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 generating set for representative 2-cocycles and n-cocycles over G, respectively. These computations have led to the finding of new cocyclic Hadamard matrices (Álvarez et al. in 2006).

[1]  Ronald Brown,et al.  The twisted Eilenberg-Zilber Theorem , 2009 .

[2]  R. J. Wilson.  Analysis situs , 1985 .

[3]  D. Flannery,et al.  Calculation of cocyclic matrices , 1996 .

[4]  J. Huebschmann.  Cohomology of nilpotent groups of class 2 , 1989 .

[5]  Kathy J. Horadam,et al.  A weak difference set construction for higher dimensional designs , 1993, Des. Codes Cryptogr..

[6]  L. Lambe,et al.  Applications of perturbation theory to iterated fibrations , 1987 .

[7]  Leonard Evens,et al.  Cohomology of groups , 1991, Oxford mathematical monographs.

[8]  Ana Romero,et al.  Computing the homology of groups: The geometric way , 2011, J. Symb. Comput..

[9]  H. Poincaré,et al.  On Analysis Situs , 2010 .

[10]  A. L.,et al.  PERTURBATION THEORY IN DIFFERENTIAL HOMOLOGICAL ALGEBRA I , 2022 .

[11]  V. K. A. M. Gugenheim,et al.  Perturbation Theory in Dierential Homological Algebra II , 1989 .

[12]  Víctor Álvarez,et al.  A Mathematica Notebook for Computing the Homology of Iterated Products of Groups , 2006, ICMS.

[13]  T. Brady.  Free resolutions for semi-direct products , 1993 .

[14]  Johannes Huebschmann,et al.  Small models for chain algebras , 1991 .

[15]  Larry A. Lambe,et al.  Computing Resolutions Over Finite p-Groups , 2001 .

[16]  Pedro Real,et al.  Homological perturbation theory and associativity , 2000 .

[17]  Saunders MacLane,et al.  On the Groups H(Π, n), II: Methods of Computation , 1954 .

[18]  J. Huebschmann.  Cohomology of metacyclic groups , 1991 .

[19]  Ana Romero,et al.  Interoperating between computer algebra systems: computing homology of groups with kenzo and GAP , 2009, ISSAC '09.

[20]  Francis Sergeraert,et al.  The Computability Problem in Algebraic Topology , 1994 .

[21]  L. Lambe.  Resolutions which split off of the bar construction , 1993 .

[22]  S. Agaian.  Hadamard Matrices and Their Applications , 1985 .

[23]  V. K. A. M. Gugenheim,et al.  On the chain-complex of a fibration , 1972 .

[24]  Samuel Eilenberg,et al.  On the Groups H(Π, n), I , 1953 .

[25]  K. J. Horadam,et al.  Generation of Cocyclic Hadamard Matrices , 1995 .

[26]  Jon P. May.  Simplicial objects in algebraic topology , 1993 .

[27]  K. J. Horadam,et al.  Cocyclic Development of Designs , 1993 .

[28]  Samuel Eilenberg,et al.  On Products of Complexes , 1953 .

[29]  Víctor Álvarez,et al.  The homological reduction method for computing cocyclic Hadamard matrices , 2009, J. Symb. Comput..

[30]  Weishu Shih,et al.  Homologie des espaces fibrés , 1962 .

[31]  Víctor Álvarez,et al.  Calculating Cocyclic Hadamard Matrices in Mathematica: Exhaustive and Heuristic Searches , 2006, ICMS.