A strong equivariant deformation retraction from the homeomorphism group of the projective plane to the special orthogonal group

Abstract

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.