E_7, 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 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.

32 pages

References in corpus (2)

• Small values of Lusternik-Schnirelmann and systolic categories for manifolds
• On manifolds satisfying stable systolic inequalities

Cited by in corpus (2)

• On manifolds satisfying stable systolic inequalities
• Systolic inequalities and Massey products in simply-connected manifolds