On infinite deficiency in $\textbf {R}^ \infty$-manifolds

Abstract:

Using the notion of inductive proper $q - 1 - {\text {LCC}}$ introduced in this note, we will prove the following theorems. Theorem 1. Let $M$ be an ${R^\infty }$-manifold and let $H:X \times I \to M$ be a homotopy such that ${H_0}$ and ${H_1}$ are ${R^\infty }$-deficient embeddings. Then, there is a homeomorphism $F$ of $M$ such that $F \circ {H_0} = {H_1}$. Moreover, if $H$ is limited by an open cover $\alpha$ of $M$ and is stationary on a closed subset ${X_0}$ of $X$ and ${W_0}$ is an open neighborhood of \[ H[(X - {X_0}) \times I] \quad {in\;M,} \] then we can choose $F$ to also be $\operatorname {St}^4(\alpha )$-close to the identity and to be the identity on $\dot X_{0} \cup (M - {W_0})$. Theorem 2. Every closed, locally ${R^\infty }({Q^\infty })$-deficient subset of an ${R^\infty }({Q^\infty })$-manifold $M$ is ${R^\infty }({Q^\infty })$-deficient in $M$. Consequently, every closed, locally compact subset of $M$ is ${R^\infty }({Q^\infty })$-deficient in $M$.