Characterizing non-separable sigma-locally compact infinite-dimensional manifolds and its applications

Abstract

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 $\ell_2^f(\tau)$.