  Topological properties of the spectrum of shallow-water waves on a rotating
spherical body are established. Particular attention is paid to its spectral
flow, i.e. the modes whose frequencies transit between the Rossby and
inertia-gravity wavebands as the zonal wave number is varied. Organising the
modes according to the number of zeros of their meridional velocity, we
conclude that the net number of modes transiting between the shallow-water
wavebands on the sphere is null, in contrast with the Matsuno spectrum. This
difference can be explained by a miscount of zeros under the $\beta$-plane
approximation. We corroborate this result with the analysis of Delplace et al
(2017) by showing that the curved metric discloses a pair of degeneracy points
in the Weyl symbol of the wave operator, non-existent under the $\beta$-plane
approximation, each of them bearing a Chern number $-1$.
