  We investigate how nontrivial topology affects the entanglement dynamics
between a detector and a quantum field and between two detectors mediated by a
quantum field. Nontrivial topology refers to both that of the base space and
that of the bundle. Using a derivative-coupling Unruh-DeWitt-like detector
model interacting with a quantum scalar field in an Einstein cylinder S1
(space) x R1 (time), we see the beating behaviors in the dynamics of the
detector-field entanglement and the detector-detector entanglement, which
distinguish from the results in the non-compact (1+1) dimensional Minkowski
space. The beat patterns of entanglement dynamics in a normal and a twisted
field with the same parameter values are different because of the difference in
the spectrum of the field modes. In terms of the kinetic momentum of the
detectors, we find that the contribution by the zero mode in a normal field to
entanglement dynamics has no qualitative difference from those by the nonzero
modes.
