Homeomorphism killing rays (Q2786845)

In the paper under review the authors deal with the following question:NEWLINENEWLINESuppose that \(Z_1\) and \(Z_2\) are compactifications of the ray \([0,\infty)\), both with remainder \(X\), under which conditions a homeomorphism of \(X\) can be extended to a homeomorphism from \(Z_1\) onto \(Z_2\)?NEWLINENEWLINEThe answer to this question is, in general, negative. One extreme case was shown by \textit{M. M. Awartani} [Topology Proc. 11, No. 2, 225--238 (1986; Zbl 0641.54031)] who proved that there is a compactification \(Z\) of the ray \([0,\infty)\), with remainder the arc \([0,1]\) such that the only homeomorphism of \([0,1]\) that can be extended to a homeomorphism of \(Z\) is the identity.NEWLINENEWLINEIn the paper under review the authors extend Awartani's result by showing that for each metric continuum \(X\), there exists a compactification \(Z\) of the ray \([0,\infty)\), with remainder \(X\) such that the only homeomorphism of \(X\) that can be extended to a homeomorphism of \(Z\) is the identity.NEWLINENEWLINEThis result is one important step in the understanding of the behavior of the extension of homeomorphisms from the remainder of compactifications of \([0,\infty)\). The authors mention some interesting problems on this area that remain unsolved. Namely:NEWLINENEWLINEQuestion 1.1. Is there a \(1/3\)-homogeneous (this means, that the number of orbits under the group of homeomorphisms is exactly \(3\)) compactification of \([0,\infty)\) with the pseudo-arc \(P\) as the remainder? In particular, is there a compactification \(Z\) of \([0,\infty)\) with the pseudo-arc \(P\) as the remainder such that each homeomorphism of \(P\) onto itself can be extended to a homeomorphism of \(Z\)?NEWLINENEWLINEQuestion 1.3. Given a homeomorphism \(h\) of the pseudo-arc \(P\), is there a compactification \(Z_h\) of \([0,\infty)\) with \(P\) as the remainder such that \(h\) can be extended to a homeomorphism of \(Z_h\)?