AR-maps obtained from cell-like maps

Abstract:

The recent solution by J. van Mill of a problem of Borsuk involves using a convexification procedure in order to produce a map $f$ from the Hilbert cube $Q$ to a non-$AR$ $X$ so that each point-inverse ${f^{ - 1}}(x)$ is a Hilbert cube. A different method of obtaining $AR$-maps from cell-like maps is described and is used to show that if there is a dimension raising cell-like map, then there is an integer $n$ and a map $f$ from $Q$ to a non-$AR$ $X$ so that each point-inverse ${f^{ - 1}}(x)$ is an $n$-cell or a point.