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October, 2014 Characterizing non-separable sigma-locally compact infinite-dimensional manifolds and its applications
Katsuhisa KOSHINO
J. Math. Soc. Japan 66(4): 1155-1189 (October, 2014). DOI: 10.2969/jmsj/06641155


For an infinite cardinal $\tau$, let $\ell_2^f(\tau)$ be the linear span of the canonical orthonormal basis of the Hilbert space $\ell_2(\tau)$ of weight $= \tau$. In this paper, we give characterizations of topological manifolds modeled on $\ell_2^f(\tau)$ and $\ell_2^f(\tau) \times \bm{Q}$, where $\bm{Q} = [-1,1]^{\mathbb{N}}$ is the Hilbert cube. We denote the full simplicial complex of cardinality $= \tau$ and the hedgehog of weight $= \tau$ by $\Delta(\tau)$ and $J(\tau)$, respectively. Using our characterization of $\ell_2^f(\tau)$, we prove that both the metric polyhedron of $\Delta(\tau)$ and the space

$J(\tau)^{\mathbb{N}}_f = \{x \in J(\tau)^{\mathbb{N}} \mid x(n) = 0 \text{ except for finitely many } n \in \mathbb{N}\}$

are homeomorphic to $\ell_2^f(\tau)$.


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Katsuhisa KOSHINO. "Characterizing non-separable sigma-locally compact infinite-dimensional manifolds and its applications." J. Math. Soc. Japan 66 (4) 1155 - 1189, October, 2014.


Published: October, 2014
First available in Project Euclid: 23 October 2014

zbMATH: 1362.57031
MathSciNet: MR3272596
Digital Object Identifier: 10.2969/jmsj/06641155

Primary: ‎57N20‎
Secondary: 54F65 , 57N17 , 57Q05 , 57Q15

Keywords: $(\ell_2(\tau) \times \bm{Q}, \ell_2^f(\tau) \times \bm{Q})$-manifold pair , $(\ell_2(\tau),\ell_2^f(\tau))$-manifold pair , $(\ell_2^f(\tau) \times \bm{Q})$-manifold , $\ell_2^f(\tau)$-manifold , $Z$-embedding , (strong) $Z$-set , ANR , full simplicial complex , hedgehog , the $\tau$-discrete $n$-cells property , the $\tau$-discrete approximation property , the strong universality

Rights: Copyright © 2014 Mathematical Society of Japan


Vol.66 • No. 4 • October, 2014
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