Fixed points of homeomorphisms and basically disconnected groups - Reznichenko, Evgenii; Sipacheva, Ol'ga (Topology Proceedings, 2024)
It is proved that if $X$ is a Tychonoff space with $\operatorname{dim}_0 X<∞, h: X → X$ is a fixed-point-free homeomorphism, and there exists a coarser paracompact topology on $X$ with respect to which $h$ remains a homeomorphism, then the extension $β h$ of $h$ to $β X$ is fixed-point-free. Consequences for topological groups are derived. In particular, it is proved that any finite-dimensional $F$-group of countable pseudocharacter contains an open Boolean subgroup and that the existence of an $ω$-representable basically disconnected group not being a $P$-space is equivalent to the existence of a nondiscrete Boolean basically disconnected group of countable pseudocharacter.