In recent years, topology gained a central role in physics. We learnt that energetics could be often explained better by classes of objects defined by having qualitative differences. In today's jargon, we say they are topologically distinct. The process of creation and annihilation of physical objects with topological charge has peculiar aspects, because it cannot proceed by continuous deformations. The consequent protection of these physical states against internal and external perturbations can therefore be striking. In condensed matter, topologically-protected states are found in electronic systems in solids, spin textures, and particle or quasiparticle wave functions. Because of their general reluctance to change topological state,methods to manipulate their topological charge should be thoroughly designed.

In this dissertation, I illustrate dynamical manipulation of a topological charge in two distinct -yet connected- research areas: nanoscopic spin textures and free electrons. In both cases, the manipulation is performed via the interaction of ultrashort light packets with the system under investigation. In the first, topologically nontrivial spin vortices, called skyrmions, are injected and controlled by the laser pulses. In FeGe, Bloch-type skyrmions are investigated via static, in-situ, and time-resolved Lorentz electronmicroscopy. Spin supercooling has been found responsible for the creation and stabilization of skyrmions in large regions of the phase diagram, controlling the amount of topological charge in the material. The cooling rate ensuing from the excitation of thematerial with strong laser pulses can reach tremendous values. This work enabled an estimate of the speed limit for laser-induced writing and erasing of such skyrmionic states.

In the second case, circularly-polarized light is used to convert an electron plane wave into an electron vortex with nonzero topological charge. A chiral plasmon photoinduced at the edges of a nanohole in a metal transfers the vortex phase structure to the electron beam via stimulated plasmon absorption and emission. In a wider picture, the precise tailoring of the free electron wave function can be obtained by electron-photon inelastic scattering in proximity of a physical object, as extensively discussed in this thesis. The new awareness of our tools to control matter waves led to the implementation of an innovative type of time-domain holography, which can be used to image nanoscopic, propagating electromagnetic fields.

Besides the single results exemplified in the different chapters, such as the topological-charge manipulation studies, this thesis contributes to the refinement of time-resolved microscopy with ultrafast electrons for the investigation of magnetic, electronic, and structural degrees of freedom. Ideas for further original experiments that could push the discussed physics to new boundaries are debated throughout the chapters.