E7, Wirtinger inequalities, Cayley 4-form, and homotopy

Abstract

We study optimal curvature-free inequalities of the type discovered by C. Loewner and M. Gromov, using a generalization of the Wirtinger inequality for the comass. Using a model for the classifying space ${\mathrm{BS}}^3$ built inductively out of ${\mathrm{BS}}^1$, we prove that the symmetric metrics of certain two-point homogeneous manifolds turn out not to be the systolically optimal metrics on those manifolds. We point out the unexpected role played by the exceptional Lie algebra $E_7$ in systolic geometry, via the calculation of Wirtinger constants. Using a technique of pullback with controlled systolic ratio, we calculate the optimal systolic ratio of the quaternionic projective plane, modulo the existence of a Joyce manifold with Spin($7$)-holonomy and unit middle-dimensional Betti number