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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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On infinite deficiency in $\textbf {R}^ \infty$-manifolds
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by Vo Thanh Liem PDF
Trans. Amer. Math. Soc. 288 (1985), 205-226 Request permission

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 $M$.
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Additional Information
  • © Copyright 1985 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 288 (1985), 205-226
  • MSC: Primary 57N20; Secondary 57N35, 58B05
  • DOI: https://doi.org/10.1090/S0002-9947-1985-0773057-2
  • MathSciNet review: 773057