Extension properties of Stone-Čech coronas and proper absolute extensors

Abstract

We characterize, in terms of $X$, extensional dimension of the Stone-Čech corona $βX \setminus X$ of locally compact and Lindelöf space $X$. The non-Lindelöf case case is also settled in terms of extending proper maps with values in $I^τ\setminus L$, where $L$ is a finite complex. Further, for a finite complex $L$, an uncountable cardinal $τ$ and a $Z_τ$-set $X$ in the Tychonov cube $I^τ$ we find necessary and sufficient condition, in terms of $I^τ\setminus X$, for $X$ to be in the class $\operatorname{AE}([L])$. We also introduce a concept of a proper absolute extensor and characterize the product $[0,1)\times I^τ$ as the only locally compact and Lindelöf proper absolute extensor of weight $τ> ω$ which has the same pseudocharacter at each point.