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

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 ${R^\infty }$.