Characterizing Manifolds Modeled on Certain Dense Subspaces of Non-Separable Hilbert Spaces

Abstract

For an infinite set $\Gamma$, let $\ell_{2}^{f}(\Gamma)$ be the linear span of the canonical orthonormal basis of the Hilbert space $\ell_{2}(\Gamma)$, that is, \[ \ell_{2}^{f}(\Gamma) = \{ x\in\ell_{2}(\Gamma) \mid x(\gamma) = 0 \text{ except for finitely many } \gamma\in\Gamma \}.\] We denote $\ell_{2}^{f}=\ell_{2}^{f}(N)$. Let $Q=[-1,1]^{\omega}$ be the Hilbert cube. In this paper, we give characterizations of manifold modeled on the following spaces: $\ell_{2}(\Gamma) \times \ell_{2}^{f}$, $\ell_2^f(\Gamma)$ and $\ell_{2}^{f}(\Gamma)\times Q$, where $\ell_{2}(\Gamma)\times \ell_{2}$ and $\ell_{2}(\Gamma)\times Q$ are homeomorphic to $\ell_{2}(\Gamma)$. Our results are obtained by suitable alteration and modification of the separable case due to Bestvina and Mogilski.