  We explore the interplay between topologies in the momentum and real spaces
to formulate a thermodynamic description of nonequilibrium injection of
topological charges under external bias. We show that the edge modes engendered
by the momentum-space topology can play a functional role of connecting the
external reservoirs to the bulk transport of topological charges in the real
space. We illustrate our general results with two examples: the spin-torque
injection of skyrmions in an electrically-biased integer quantum Hall system,
and the vortex injection in a topological $p\,+\,i\,p$ superconductor coupled
to heat reservoirs. Based on the universal fractional entropy of the Majorana
zero modes bound to the vortices, their controllable injection proposed in this
work could provide a route for creating and manipulating Majorana fermions.
