  In spite of flourishing studies on the topology of spin waves, a generic
framework to classify and compute magnon band topology in non-collinear magnets
is still missing. In this work we provide such a theory framework, by mapping
an arbitrary linear spin wave into a local free-fermion Hamiltonian with
exactly the same spectrum, symmetry implementation and band topology, which
allows for a full classification and calculation on any topological properties
of magnon bands. We apply this fermionization approach to honeycomb Kitaev
magnet $\alpha$-RuCl$_3$, and show the existence of topologically protected
magnon band crossings, and field-induced magnon Chern bands under small
magnetic fields.
