Matroids and quotients of spheres

Abstract.

For any linear quotient of a sphere, $X=S^{n-1}/\Gamma,$ where $\Gamma$ is an elementary abelian p–group, there is a corresponding ${\mathbb F}_p$ representable matroid $M_X$ which only depends on the isometry class of X. When p is 2 or 3 this correspondence induces a bijection between isometry classes of linear quotients of spheres by elementary abelian p–groups, and matroids representable over $\F_p.$ Not only do the matroids give a great deal of information about the geometry and topology of the quotient spaces, but the topology of the quotient spaces point to new insights into some familiar matroid invariants. These include a generalization of the Crapo–Rota critical problem inequality $\chi(M;p^k)\ge 0,$ and an unexpected relationship between $\mu(M)$ and whether or not the matroid is affine.