  This is the third paper in a series on oriented matroids and Grassmannians.
We construct a $(\mathrm{O}_3\times\mathbb{Z}_2)$-equivariant strong
deformation retraction from the homeomorphism group of the 2-sphere to
$\mathrm{O}_3$, where the action of $\mathbb{Z}_2$ is generated by antipodal
reflection acting on the right, and $\mathrm{O}_3$ acts on the left by
isometry. Quotienting by the antipodal map induces a
$\mathrm{SO}_3$-equivariant strong deformation retraction from the
homeomorphism group of the projective plane to $\mathrm{SO}_3$. The same holds
for subgroups of homeomorphisms that preserve the system of null sets. This
confirms a conjecture of Mary-Elizabeth Hamstrom.
