On the homology of the harmonic archipelago
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?}
scientific article; zbMATH DE number 6039992