There was movement that was stationary, for the four-velocity had passed
  around
  Is the Doppler interpretation of galaxy redshifts in a
Friedmann-Lemaitre-Robertson-Walker (FLRW) model valid in the context of the
approach to comoving spatial sections pioneered by de Sitter, Friedmann,
Lemaitre and Robertson, i.e. according to which the 3-manifold of comoving
space is characterised by both its curvature and topology? Holonomy
transformations for flat, spherical and hyperbolic FLRW spatial sections are
proposed. By quotienting a simply-connected FLRW spatial section by an
appropriate group of holonomy transformations, the Doppler interpretation in a
non-expanding Minkowski space-time, obtained via four-velocity parallel
transport along a photon path, is found to imply that an inertial observer is
receding from herself at a speed greater than zero, implying contradictory
world-lines. The contradiction in the multiply-connected case occurs for
arbitrary redshifts in the flat and spherical cases, and for certain large
redshifts in the hyperbolic case. The link between the Doppler interpretation
of redshifts and cosmic topology can be understood physically as the link
between parallel transport along a photon path and the fact that the comoving
spatial geodesic corresponding to a photon's path can be a closed loop in an
FLRW model of any curvature. Closed comoving spatial loops are fundamental to
cosmic topology.
