Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



On infinite deficiency in $ {\bf R}\sp \infty$-manifolds

Author: Vo Thanh Liem
Journal: Trans. Amer. Math. Soc. 288 (1985), 205-226
MSC: Primary 57N20; Secondary 57N35, 58B05
MathSciNet review: 773057
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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

$\displaystyle 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$.

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Keywords: Isotopy, collar, bicollar, $ \alpha $-limited, direct limit space, $ 1{\text{-LCC}}$ embedding, unknotting theorem
Article copyright: © Copyright 1985 American Mathematical Society