The topological phase acquired by vector vortex optical beams is investigated. Under local unitary operations on their polarization and transverse degrees of freedom, the vector vortices can only acquire discrete geometric phase values, 0 or $π\,$, associated with closed paths belonging to different homotopy classes on the SO(3) manifold. These discrete values are demonstrated through interferometric measurements and the spin-orbit mode separability is associated to the visibility of the interference patterns. The local unitary operations performed on the vector vortices involved both polarization and transverse mode transformations with birefringent wave plates and astigmatic mode converters. The experimental results agree with our theoretical simulations and generalize our previous results obtained with polarization transformations only.

6 pages, 5 figures, submitted to Physical Review A