For exact (i.e., non-paraxial) waves ψ representing freely propagating Gaussian beams in two and three dimensions, the patterns of phase singularities, that is zeros of ψ, are studied in detail. The zeros (points in two dimensions, and rings in three) are phase dislocations (optical vortices). The waves depend on a single parameter L, representing the radius of the waist of the beam. As L increases, pairs of dislocations interact and depart from the focal plane. Each such interaction comprises three events where the phase topology of ψ changes; each event is a reaction between the dislocations and associated phase saddles, conserving two topological quantum numbers. The same behaviour was predicted and observed by Karman et al. for beams truncated by apertures. The geometrical sensitivity of the wave to L is astonishing: changes in phase topology can occur when the waist expands by a few thousandths of a wavelength. The integral representing ψ is evaluated asymptotically, leading to a global explanation of the dislocations and topological events in terms of interference between complex and diffracted rays. Locally, all details of the topological changes can be captured by a local model, constructed using a classification devised by Nye.