Duke Mathematical Journal
- Duke Math. J.
- Volume 146, Number 1 (2009), 35-70.
$E_7$, Wirtinger inequalities, Cayley $4$-form, and homotopy
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 $BS^3$ 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
Duke Math. J. Volume 146, Number 1 (2009), 35-70.
First available in Project Euclid: 17 December 2008
Permanent link to this document
Digital Object Identifier
Zentralblatt MATH identifier
Primary: 53C23: Global geometric and topological methods (à la Gromov); differential geometric analysis on metric spaces
Secondary: 55R37: Maps between classifying spaces 17B25: Exceptional (super)algebras
Bangert, Victor; Katz, Mikhail G.; Shnider, Steven; Weinberger, Shmuel. E 7 , Wirtinger inequalities, Cayley 4 -form, and homotopy. Duke Math. J. 146 (2009), no. 1, 35--70. doi:10.1215/00127094-2008-061. http://projecteuclid.org/euclid.dmj/1229530284.