  The nontrivial topology of spin systems such as skyrmions in real space can
promote complex electronic states. Here, we provide a general viewpoint at the
emergence of topological electronic states in spin systems based on the methods
of noncommutative K-theory. By realizing that the structure of the observable
algebra of spin textures is determined by the algebraic properties of the
noncommutative hypertorus, we arrive at a unified understanding of topological
electronic states which we predict to arise in various noncollinear setups. The
power of our approach lies in an ability to categorize emergent topological
states algebraically without referring to smooth real- or reciprocal-space
quantities. This opens a way towards an educated design of topological phases
in aperiodic, disordered, or non-smooth textures of spins and charges
containing topological defects.
