  We study the linear response of four dimensional magnetized black holes and
regular topological stars arising from dimensional compactification of
Einstein-Maxwell theory in five dimensions. We consider both radial and
nonradial perturbations and study the stability of these solutions, both in the
frequency and in the time domain. Due to the presence of magnetic fluxes in the
background, axial (i.e., odd-parity) gravitational perturbations are coupled to
polar (i.e., even-parity) electromagnetic perturbations (Type-I sector) whereas
polar gravitational and scalar perturbations are coupled to axial
electromagnetic ones (Type-II sector). We provide a comprehensive analytical
and numerical study of the radial perturbations and of the Type I sector,
finding no evidence of linear instabilities (besides the already known
Gregory-Laflamme instability of black strings occurring only in a certain range
of the parameters), even despite the fact that the effective potential for
radial perturbations of topological stars is negative and divergent near the
inner boundary. Ultracompact topological stars exhibit long-lived trapped modes
that give rise to echoes in the time-domain response. Their prompt response is
very similar to that of the corresponding black hole with comparable
charge-to-mass ratio. This provides a concrete realization of ultracompact
objects arising from a well-defined theory. The numerical analysis of the
Type-II sector will appear in a companion paper.
