Logo image
Back
The diagonal comultiplication on homology
Journal article   Peer reviewed

The diagonal comultiplication on homology

K.J. Horadam
Journal of Pure and Applied Algebra, Vol.20(2), pp.165-172
1981
DOI: https://doi.org/10.1016/0022-4049(81)90090-6
url
Free to Read *No subscription requiredView

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.

Details

Metrics

52 Record Views
2 Times Cited - Web of Science

InCites Highlights

These are selected metrics from InCites Benchmarking & Analytics tool, related to this output

Citation topics
9 Mathematics
9.28 Pure Maths
9.28.500 Geometric Group Theory
Web Of Science research areas
Mathematics
Mathematics, Applied
ESI research areas
Mathematics
Logo image