  We present integral-field spectroscopy of 27 galaxies in the Coma cluster
observed with the Oxford SWIFT spectrograph, exploring the kinematic
morphology-density relationship in a cluster environment richer and denser than
any in the ATLAS3D survey. Our new data enables comparison of the kinematic
morphology relation in three very different clusters (Virgo, Coma and Abell
1689) as well as to the field/group environment. The Coma sample was selected
to match the parent luminosity and ellipticity distributions of the early-type
population within a radius 15' (0.43 Mpc) of the cluster centre, and is limited
to r' = 16 mag (equivalent to M_K = -21.5 mag), sampling one third of that
population. From analysis of the lambda-ellipticity diagram, we find 15+-6% of
early-type galaxies are slow rotators; this is identical to the fraction found
in the field and the average fraction in the Virgo cluster, based on the
ATLAS3D data. It is also identical to the average fraction found recently in
Abell 1689 by D'Eugenio et al.. Thus it appears that the average slow rotator
fraction of early type galaxies remains remarkably constant across many
different environments, spanning five orders of magnitude in galaxy number
density. However, within each cluster the slow rotators are generally found in
regions of higher projected density, possibly as a result of mass segregation
by dynamical friction. These results provide firm constraints on the mechanisms
that produce early-type galaxies: they must maintain a fixed ratio between the
number of fast rotators and slow rotators while also allowing the total
early-type fraction to increase in clusters relative to the field. A complete
survey of Coma, sampling hundreds rather than tens of galaxies, could probe a
more representative volume of Coma and provide significantly stronger
constraints, particularly on how the slow rotator fraction varies at larger
radii.
