  We recently showed that spin fluctuations of noncoplanar magnetic states can
induce topological superconductivity in an adjacent normal metal [M{\ae}land et
al., Phys. Rev. Lett. 130, 156002 (2023)]. The noncolinear nature of the spins
was found to be essential for this result, while the necessity of noncoplanar
spins was unclear. In this paper we show that magnons in coplanar, noncolinear
magnetic states can mediate topological superconductivity in a normal metal.
Two models of the Dzyaloshinskii-Moriya interaction are studied to illustrate
the need for a sufficiently complicated Hamiltonian describing the magnetic
insulator. The Hamiltonian, in particular the specific form of the
Dzyaloshinskii-Moriya interaction, affects the magnons and by extension the
effective electron-electron interaction in the normal metal. Symmetry arguments
are applied to complement this discussion. We solve a linearized gap equation
in the case of weak-coupling superconductivity. The result is a
time-reversal-symmetric topological superconductor, as confirmed by calculating
the topological invariant. In analogy with magnon-mediated superconductivity
from antiferromagnets, Umklapp scattering enhances the critical temperature of
superconductivity for certain Fermi momenta.
