Homology, Homotopy and Applications

Volume 20 (2018)

Number 2

Distributed computation of low-dimensional cup products

Pages: 41 – 59

DOI: http://dx.doi.org/10.4310/HHA.2018.v20.n2.a3

Authors

Nisreen Alokbi (School of Mathematics, National University of Ireland, Galway, Ireland; and Department of Mathematics, College of Computer Science and Mathematics, University of Wasit, Iraq)

Graham Ellis (School of Mathematics, National University of Ireland, Galway, Ireland)

Abstract

We describe a distributed algorithm for computing the cup product $\cup : H^1 (X, \mathbb{Z}) \times H^1 (X, \mathbb{Z}) \to H^2(X, \mathbb{Z})$ on the cohomology of a finite regular CW-space. A serial implementation of the algorithm is illustrated in two applied topological settings: (i) 3-dimensional digital images; (ii) topological data analysis of a finite sample of points from a metric space. For the second of these illustrations we introduce a cohomological enrichment of the Mapper clustering procedure which may be of independent interest.

Keywords

cohomology, cup product, digital image, data analysis

2010 Mathematics Subject Classification

55N99

Received 8 May 2017

Received revised 16 December 2017

Published 18 April 2018