A process and apparatus to make an edge or a collar featuring a complex structure on extruded, deep-drawn and deep-drawn/wire-drawn metal rough pieces with which bottles for the beverage and food sector or for technical use are particularly obtained, suitable for the application of a closing cap and fit to use on a tapering machine. The process includes a starting operating stage to deform an upper end portion of a metal holder by turning it outwardly so as to obtain an edge or a collar with a basically circular section, one or more intermediate operating stages to deform the edge or collar by squeezing in the radial direction and stretching towards the bottom of the metal holder, a final operating deformation stage to obtain an edge or collar with a complex ovoidal profile for the application of at least two caps of different types.