## 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