Duke Mathematical Journal

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

Victor Bangert, Mikhail G. Katz, Steven Shnider, and Shmuel Weinberger

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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 BS3 built inductively out of BS1, 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 E7 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

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Duke Math. J., Volume 146, Number 1 (2009), 35-70.

First available in Project Euclid: 17 December 2008

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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. https://projecteuclid.org/euclid.dmj/1229530284

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