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An unknotting theorem in $ Q\sp{\infty }$-manifolds


Author: Vo Thanh Liem
Journal: Proc. Amer. Math. Soc. 82 (1981), 125-132
MSC: Primary 57N20; Secondary 57N37
DOI: https://doi.org/10.1090/S0002-9939-1981-0603615-9
MathSciNet review: 603615
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Abstract: In this note, we prove the following unknotting theorem.

Theorem. Let $ M$ be a $ {Q^\infty }$-manifold and let $ F:X \times I \to M$ be a homotopy such that $ {F_0}$ and $ {F_1}$ are $ {Q^\infty }$-deficient embeddings. Then, there is an isotopy $ H:M \times I \to M$ such that $ {H_0} = {\text{id}}$ and $ {H_1} \circ {F_0} = {F_1}$. Moreover, if $ F$ is limited by an open cover $ \alpha $ of $ M$ and is stationary on a closed subset $ {X_0}$ of $ X$, then we may choose $ H$ to also be limited by $ {\text{S}}{{\text{t}}^4}(\alpha )$ and to be the identity on $ F({X_0} \times I)$.

However, a similar unknotting theorem for $ Z$-embeddings does not hold true in $ {Q^\infty }$ and $ {R^\infty }$.


References [Enhancements On Off] (What's this?)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1981-0603615-9
Keywords: Hilbert cube, direct limit space, $ Z$-set, inductive $ Z$-set, $ {Q^\infty }$-deficient, isotopy, unknotting theorem
Article copyright: © Copyright 1981 American Mathematical Society

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