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

TL;DR

This paper constructs an equivariant deformation retraction from the homeomorphism group of the projective plane to the special orthogonal group, confirming a conjecture and extending previous results on the sphere.

Contribution

It provides a new equivariant deformation retraction from the homeomorphism group of the projective plane to SO(3), confirming Hamstrom's conjecture.

Findings

Constructs a strong deformation retraction from the homeomorphism group of the sphere to O(3).

Induces a deformation retraction from the homeomorphism group of the projective plane to SO(3).

Extends the result to subgroups preserving null sets.

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.