Get 20M+ Full-Text Papers For Less Than $1.50/day. Start a 14-Day Trial for You or Your Team.

Learn More →

Harmonic representatives in homology over arbitrary fields

Harmonic representatives in homology over arbitrary fields We introduce a notion of harmonic chain for chain complexes over fields of positive characteristic. A list of conditions for when a Hodge decomposition theorem holds in this setting is given and we apply this theory to finite CW complexes. An explicit construction of the harmonic chain within a homology class is described when applicable. We show how the coefficients of usual discrete harmonic chains due to Eckmann can be reduced to localizations of the integers, allowing us to compare classical harmonicity with the notion introduced here. We focus on applications throughout, including CW decompositions of orientable surfaces and examples of spaces arising from sampled data sets. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Journal of Applied and Computational Topology Springer Journals

Harmonic representatives in homology over arbitrary fields

Download PDF
28 pages

Loading next page...
 
/lp/springer-journals/harmonic-representatives-in-homology-over-arbitrary-fields-Zi6byVSt0I
Publisher
Springer Journals
Copyright
Copyright © The Author(s), under exclusive licence to Springer Nature Switzerland AG 2023. Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
ISSN
2367-1726
eISSN
2367-1734
DOI
10.1007/s41468-023-00117-w
Publisher site
See Article on Publisher Site

Abstract

We introduce a notion of harmonic chain for chain complexes over fields of positive characteristic. A list of conditions for when a Hodge decomposition theorem holds in this setting is given and we apply this theory to finite CW complexes. An explicit construction of the harmonic chain within a homology class is described when applicable. We show how the coefficients of usual discrete harmonic chains due to Eckmann can be reduced to localizations of the integers, allowing us to compare classical harmonicity with the notion introduced here. We focus on applications throughout, including CW decompositions of orientable surfaces and examples of spaces arising from sampled data sets.

Journal

Journal of Applied and Computational Topology – Springer Journals

Published: Mar 20, 2023

References

  • Hodge theory for combinatorial geometries
    Adiprasito, K; Huh, J; Katz, E
  • Hodge theory in combinatorics
    Baker, M
  • Ripser: efficient computation of vietoris-rips persistence barcodes
    Bauer, U
  • A geometric property of the least squares solution of linear equations
    Ben-Tal, A; Teboulle, M
  • Kirchhoff’s theorems in higher dimensions and Reidemeister torsion
    Catanzaro, MJ; Chernyak, VY; Klein, JR
  • A higher Boltzmann distribution
    Catanzaro, MJ; Chernyak, VY; Klein, JR
  • Harmonische funktionen und randwertaufgaben in einem komplex
    Eckmann, B
  • Barcodes: the persistent topology of data
    Ghrist, R
  • Statistical ranking and combinatorial Hodge theory
    Jiang, X; Lim, L-H; Yao, Y; Ye, Y
  • Can one hear the shape of a drum?
    Kac, M
  • A higher-dimensional homologically persistent skeleton
    Kališnik, S; Kurlin, V; Lešnik, D
  • Hodge laplacians on graphs
    Lim, L-H
  • Random walks on simplicial complexes and harmonics
    Mukherjee, S; Steenbergen, J
  • Generalized inverses of matrices with entries taken from an arbitrary field
    Pearl, MH
  • A generalized inverse for matrices
    Penrose, R
  • Spectral ranking
    Vigna, S
  • Persistent spectral graph
    Wang, R; Nguyen, DD; Wei, G-W