E_7, Wirtinger inequalities, Cayley 4-form, and homotopy

TL;DR

This paper explores optimal curvature inequalities and systolic geometry, revealing the role of E_7 Lie algebra and calculating systolic ratios for specific manifolds using advanced geometric techniques.

Contribution

It introduces a generalized Wirtinger inequality, links E_7 to systolic geometry, and computes optimal systolic ratios for quaternionic projective planes.

Findings

Symmetric metrics are not systolically optimal on certain two-point homogeneous manifolds.

E_7 plays an unexpected role in systolic geometry through Wirtinger constants.

Calculated the optimal systolic ratio of the quaternionic projective plane.

Abstract

We study optimal curvature-free inequalities of the type discovered by C. Loewner and M. Gromov, using a generalisation of the Wirtinger inequality for the comass. Using a model for the classifying space BS^3 built inductively out of 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.