Products of conjugacy classes in SU(2)

We obtain a complete description of collections of n conjugacy classes in SU(2) with the property that the multiplication map from the product of these n conjugacy classes to SU(2) is surjective. The basic instrument is a characterization of collections of n+1 conjugacy classes in SU(2) such that the image of the multiplication map from these conjugacy classes to SU(2) contains the identity matrix, which generalizes a result of Jeffrey and Weitsman.