A process to make an edge or a collar (18') 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, this process including, in a sequence, a starting operating stage meant to deform an upper end portion (12) of a metal holder (10) by turning it outwardly so as to obtain an edge or a collar (18) with a basically circular section, one or more intermediate operating stages fit to deform an edge or collar (180 by its squeezing in the radial direction and its stretching towards the bottom of the metal holder (10), a final operating deformation stage and possibly a surface finishing stage fit to obtain an edge or collar (18') with a complex ovoidal profile for the application of at least two caps of different types, a possible working stage that is preliminary to the ones mentioned above, fit to reduce the thickness of a side surface (16) of the upper end portion (12) of the metal holder (10).