# The diagonal comultiplication on homology

Horadam, K.J.
(1981)
*The diagonal comultiplication on homology.*
Journal of Pure and Applied Algebra, 20
(2).
pp. 165-172.

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## Abstract

This paper describes the diagonal comultiplication (or cup coproduct) defined on integral homology modules of groups. Analysis of this coproduct should provide a new method of testing for non-isomorphism of groups which have isomorphic integral homology modules; here, the dimension two coproduct is applied to this problem. The first part (Section 2) is couched in terms of groupnets (Brandt groupoids) and shows two things: that there exists a cup product defined on the integral cohomology of any groupnet, extending that for groups, and that there exists a comultiplication defined on the integral homology of any group, natural up to dimension two, which gives the homology modules the structure of a commutative graded co-ring. In the second part (Sections 3 and 4), this diagonal comultiplication R is constructed to dimension two, and the information it carries about the lower central series of a group G is investigated. Modulo torsion in Hr(G; Z), Rz induces an abelian group homomorphism with cokernel GZ/G3, which distinguishes between large classes of groups, in particular the one-relator groups with non-trivial multiplicator, and the finitely-generated nilpotent groups of class two whose relators are all in the commutator subgroup.

Item Type: | Journal Article |
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Murdoch Affiliation: | School of Mathematical and Physical Sciences |

Publisher: | Elsevier B.V. |

Copyright: | © 2014 Elsevier B.V. |

URI: | http://researchrepository.murdoch.edu.au/id/eprint/21226 |

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