On $R^{\infty}$-manifolds and $Q^{\infty}$-manifolds, Ⅱ: Infinite deficiency

Abstract

We introduce the notion of $D$-sets ($D^{*}$-sets) and establish the Unknotting Theorem for $D$-sets in a manifold modeled on $R^{\infty}=$dir $\lim R^{n}$ or $Q^{\infty}=$ dir $\lim Q^{n}$, where $Q$ is the Hilbert cube. This yields equality of $D$-sets, $D^{*}$-sets and intinite (i.e.,$R^{\infty}$ or $Q^{\infty}$) deficient sets. Our Theorem corresponds to a weak version of the Unknotting Theorem for infinite deficient sets proved by V.T. Liem. However our proof is elementary and short. And we give an alternative proof of the Infinite Deficient Embedding Approximation Theorem due to Liem. Using Anderson-McCharen's trick, this Approximation Theorem strengthens our Unknotting Theorem in the strong form. Moreover, we show that the union of two $R^{\infty}$ (or $Q^{\infty}$) manifolds meeting in an $R^{\infty}$ (or $Q^{\infty}$) manifold is also an $R^{\infty}$ (or $R^{\infty}$) manifold, and that for any space $X$, $X\times R$ is an $Q^{\infty}$ (or $Q^{\infty}$) manifold if and only if so is $X\times I$.