  Understanding the interaction of real- and reciprocal space topology in
skyrmion crystals is an open problem. We approach it from the viewpoint of
$C^\ast$-algebras and calculate all admissible Chern numbers of a strongly
coupled tight-binding skyrmion system on a triangular lattice as a function of
Fermi energy and texture parameters. Our analysis reveals the topological
complexity of electronic states coupled to spin textures, and the failure of
the adiabatic picture to account for it in terms of emergent electromagnetism.
On the contrary, we explain the discontinuous jumps in the real-space winding
number in terms of collective evolution in real-, reciprocal, and mixed space
Chern numbers. Our work sets the stage for further research on topological
dynamics in complex dynamic spin textures coupled to external fields.
