We present a joint analysis of panoramic Spitzer/MIPS mid-infrared and GALEX ultraviolet imaging of the Shapley supercluster at z=0.048. Combining this with spectra of 814 supercluster members and 1.4GHz radio continuum maps, this represents the largest complete census of star-formation (both obscured and unobscured) in local cluster galaxies to date, reaching SFRs~0.02Msun/yr. We take advantage of this comprehensive panchromatic dataset to perform a detailed analysis of the nature of star formation in cluster galaxies, using several quite independent diagnostics of the quantity and intensity of star formation to develop a coherent view of the types of star formation within cluster galaxies. We observe a robust bimodality in the infrared (f_24/f_K) galaxy colours, which we are able to identify as another manifestation of the broad split into star-forming spiral and passive elliptical galaxy populations seen in UV-optical surveys. This diagnostic also allows the identification of galaxies in the process of having their star formation quenched as the infrared analogue to the UV green valley population. The bulk of supercluster galaxies on the star-forming sequence have specific-SFRs consistent with local field specific-SFR-M* relations, and form a tight FIR-radio correlation confirming that their FIR emission is due to star formation. We show that 85% of the global SFR is quiescent star formation within spiral disks, as manifest by the observed sequence in the IRX-beta relation being significantly offset from the starburst relation of Kong et al. (2004), while their FIR-radio colours indicate dust heated by low-intensity star formation. Just 15% of the global SFR is due to nuclear starbursts. The vast majority of star formation seen in cluster galaxies comes from normal infalling spirals who have yet to be affected by the cluster environment.
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