Whitney levels in hyperspaces of certain Peano continua

Abstract:

Let $X$ be a Peano continuum. Let ${2^x}$ (resp., $C(X)$) be the space of all nonempty compacta (resp., subcontinua) of $X$ with the Hausdorff matric. Let $\omega$ be a Whitney map defined on $\mathcal {H}={2^{X}}$ or $C(X)$ such that $\omega$ is admissible (this requires the existence of a certain type of deformation of $\mathcal {H}$). If $\mathcal {H}=C(X)$, assume $X$ contains no free arc. Then, for any ${t_0} \in (0,\omega (X))$, it is proved that ${\omega ^{ - 1}}({t_0}), {\omega ^{ - 1}}([0, {t_0}])$, and ${\omega ^{ - 1}}([{t_0}, \omega (X)])$ are Hilbert cubes. This is an analogue of the Curtis-Schori theorem for $\mathcal {H}$. A general result for the existance of admissible Whitney maps is proved which implies that these maps exist when $X$ is starshaped in a Banach space or when $X$ is a dendrite. Using these results it is shown, for example that being an AR, an ANR, an LC space, or an ${\text {L}}{{\text {C}}^n}$ space is not strongly Whitney-reversible.