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 DEugenio 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.
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