On the homology of the harmonic archipelago
type: Item
instance of: scholarly article
title: On the homology of the harmonic archipelago
zbMATH Open document ID: 1250.54033
author: Umed H. Karimov
author: Dušan Repovš
published in: Central European Journal of Mathematics
publication date: 2012-05-31T00:00:00Z
full work available at URL: https://arxiv.org/abs/1203.4274
review text: Consider two copies of the \textit{Hawaiian Earring} \(\mathbb{H}\) in \(\mathbb{R}^3\) as follows: \[ \mathbb{H}_{+}=\bigcup_{n\in \mathbb{N}}\{(x,y,0)\in \mathbb{R}^3|x^2+(y-1/n)^2=(1/n)^2\}, \] \[ \mathbb{H}_{-}=\bigcup_{n\in \mathbb{N}}\{(x,y,0)\in \mathbb{R}^3|x^2+(y+1/n)^2=(1/n)^2\}. \] The \textit{Griffiths space} \(\mathcal{G}\) is a one-point union of two cones over the Hawaiian Earring \(\mathbb{H}\) as follows: \[ \mathcal{G}=C(\mathbb{H}_{+},(0,0,1))\cup C(\mathbb{H}_{-}(0,0,-1)). \] The Harmonic Archipelago \(\mathcal{HA}\) is a noncompact space which is obtained by adjoining a sequence of tall disks between consecutive loops of the Hawaiian Earring. The main results of the paper under review are calculating the singular homology and Čech cohomology groups of the Harmonic Archipelago as follows: \[ H_1(\mathcal{HA};\mathbb{Z})\cong (\prod_{i=1}^{\infty}\mathbb{Z})/(\sum_{i=1}^{\infty}\mathbb{Z}),\;\;H_n(\mathcal{HA};\mathbb{Z})\cong 0,\;\mathrm{for\;all}\;n\geq 2. \] \[ \check{H}^2(\mathcal{HA};\mathbb{Z})\cong (\prod_{i=1}^{\infty}\mathbb{Z})/(\sum_{i=1}^{\infty}\mathbb{Z}),\;\;\check{H}^n(\mathcal{HA};\mathbb{Z})\cong 0,\;\mathrm{for\;all}\;n\neq 0,2. \] It is well known that the Griffiths space \(\mathcal{G}\) is a cell-like space and therefore it has Čech cohomology groups of the one-point space, \(\check{H}^*(\mathcal{G};\mathbb{Z})\cong \check{H}^*(\mathrm{pt};\mathbb{Z})\). As a consequence the authors give the following important and interesting corollary: \textit{The Griffiths space \(\mathcal{G}\) and the Harmonic Archipelago \(\mathcal{HA}\) are not homotopy equivalent.} The above result is a step towards solving the following interesting open problem: \textit{Are the fundamental groups of the Griffiths space \(\mathcal{G}\) and the Harmonic Archipelago \(\mathcal{HA}\) isomorphic?}
reviewed by: Behrooz Mashayekhy
Mathematics Subject Classification ID: 54F15
Mathematics Subject Classification ID: 55N15
Mathematics Subject Classification ID: 54G20
Mathematics Subject Classification ID: 57M05
zbMATH DE Number: 6039992
zbMATH Keywords: Griffiths space
zbMATH Keywords: Harmonic Archipelago
zbMATH Keywords: Hawaiian Earring
zbMATH Keywords: fundamental group
zbMATH Keywords: trivial shape
zbMATH Keywords: Peano continuum
zbMATH Keywords: wild topology
MaRDI profile type: Publication
OpenAlex ID: W3103417907
arXiv ID: 1203.4274
cites work: Singular homology of one-dimensional spaces
cites work: The Singular Homology of the Hawaiian Earring
cites work: Q5609457
cites work: THE FUNDAMENTAL GROUP OF TWO SPACES WITH A COMMON POINT
cites work: Q4070007
cites work: Q4819371
cites work: Archipelago groups
cites work: Q5522742
DOI: 10.2478/S11533-012-0038-2
description: scientific article; zbMATH DE number 6039992