Duke Mathematical Journal

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

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

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


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

Article information

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

Export citation


  • B. S. Acharya, On mirror symmetry for manifolds of exceptional holonomy, Nuclear Phys. B 524 (1998), 269--282.
  • J. F. Adams, Lectures on Exceptional Lie Groups, Chicago Lectures in Math., Univ. of Chicago Press, Chicago, 1996.
  • L. V. Antonyan, Classification of four-vectors of an eight-dimensional space (in Russian), Trudy Sem. Vektor. Tenzor. Anal. 20 (1981), 144--161.
  • I. K. Babenko, Asymptotic invariants of smooth manifolds, Russian Acad. Sci. Izv. Math. 41 (1993), 1--38.
  • —, Forte souplesse intersystolique de variétés fermées et de polyèdres, Ann. Inst. Fourier (Grenoble) 52 (2002), 1259--1284.
  • —, ``Géométrie systolique des variétés de groupe fondamental $\Z_2$'' in Séminaire de théorie spectrale et géométrie, Vol. 22: Année 2003--2004., Semin. Theor. Spectr. Geom. 22, Univ. Grenoble I, Institut Fourier, Saint-Martin-d'Hères, France, 2004, 25--52.
  • —, Topologie des systoles unidimensionelles, Enseign. Math. (2) 52 (2006), 109--142.
  • V. Bangert and M. Katz, Stable systolic inequalities and cohomology products, Comm. Pure Appl. Math. 56 (2003), 979--997.
  • —, An optimal Loewner-type systolic inequality and harmonic one-forms of constant norm, Comm. Anal. Geom. 12 (2004), 703--732.
  • K. Becker, M. Becker, D. R. Morrison, H. Ooguri, Y. Oz, and Z. Yin, Supersymmetric cycles in exceptional holonomy manifolds and Calabi-Yau four-folds, Nuclear Phys. B 480 (1996), 225--238.
  • M. Berger, Du côté de chez Pu, Ann. Sci. École Norm. Sup. (4) 5 (1972), 1--44.
  • —, Systoles et applications selon Gromov, Astérisque 216 (1993), 279--310., Séminaire Bourbaki 1992/1993, no. 771.
  • —, A Panoramic View of Riemannian Geometry, Springer, Berlin, 2003.
  • —, What is,$\ldots$,a systole? Notices Amer. Math. Soc. 55 (2008), 374--376.
  • J. Bochnak, M. Coste, and M.-F. Roy, Real Algebraic Geometry, Ergeb. Math. Grenzgeb. (3) 36, Springer, Berlin, 1998.
  • M. Brunnbauer, Filling inequalities do not depend on topology, to appear in J. Reine Angew. Math., preprint,\arxiv0706.2790v3[math.GT]
  • —, Homological invariance for asymptotic invariants and systolic inequalities, to appear in Geom. Funct. Anal. (2008), preprint,\arxivmath/0702789v3[math.GT]
  • —, On manifolds satisfying stable systolic inequalities, to appear in Math. Annalen, preprint,\arxiv0708.2589v2[math.GT]
  • R. L. Bryant, Metrics with exceptional holonomy, Ann. of Math. (2) 126 (1987), 525--576.
  • R. L. Bryant and S. M. Salamon, On the construction of some complete metrics with exceptional holonomy, Duke Math. J. 58 (1989), 829--850.
  • J. Cheeger and D. G. Ebin, Comparison Theorems in Riemannian Geometry, North-Holland Math. Lib. 9, North-Holland, Amsterdam, 1975.
  • J. Dadok, R. Harvey, and F. Morgan, Calibrations on $\R\sp 8$, Trans. Amer. Math. Soc. 307, no. 1 (1988), 1--40.
  • A. N. Dranishnikov, M. G. Katz and Y. B. Rudyak, Small values of the Lusternik-Schnirelman category for manifolds, Geom. Topol. 12 (2008), 1711--1728.
  • B. Eckmann, Üeber die Homotopiegruppen von Gruppenräumen, Comment. Math. Helv. 14 (1942), 234--256.
  • J. Eells Jr. and N. H. Kuiper, Manifolds which are like projective planes, Inst. Hautes Études Sci. Publ. Math. 14 (1962), 5--46.
  • D. B. A. Epstein, The degree of a map, Proc. London Math. Soc. (3) 16 (1966), 369--383.
  • H. Federer, Geometric Measure Theory, Grundlehren Math. Wiss. 153, Springer, Berlin, 1969.
  • —, Real flat chains, cochains and variational problems, Indiana Univ. Math. J. 24 (1974/75), 351--407.
  • V. Gatti [Kac] and E. Viniberghi [Vinberg], Spinors of $13$-dimensional space, Adv. in Math. 30 (1978), 137--155.
  • M. Gromov, Structures métriques pour les variétés riemanniennes, Textes Math. 1, CEDIC, Paris, 1981.
  • —, Filling Riemannian manifolds, J. Differential Geom. 18 (1983), 1--147.
  • —, ``Systoles and intersystolic inequalities'' in Actes de la table ronde de géométrie différentielle (Luminy, France, 1992), Sémin. Congr. 1, Soc. Math. France, Montrouge, 1996.
  • —, Metric Structures for Riemannian and Non-Riemannian Spaces, with appendices by M. Katz, P. Pansu, and S. Semmes, Progr. Math. 152, Birkhäuser, Boston, 1999.
  • —, Metric Structures for Riemannian and Non-Riemannian Spaces, reprint of the 2001 English ed., with appendices by M. Katz, P. Pansu, and S. Semmes, Mod. Birkhäuser Class., Birkhäuser, Boston, 2007.
  • R. Harvey and H. B. Lawson Jr. Calibrated geometries, Acta Math. 148 (1982), 47--157.
  • C. Horowitz, M. Katz, and K. Usadi Katz, Loewner's torus inequality with isosystolic defect and Liouville's equation for curvature, preprint,\arxiv0803.0690v1[math.DG]
  • G. A. Hunt, A theorem of Elie Cartan, Proc. Amer. Math. Soc. 7 (1956), 307--308.
  • D. D. Joyce, Compact Manifolds with Special Holonomy, Oxford Math. Monogr., Oxford Univ. Press, Oxford, 2000.
  • M. G. Katz, Counterexamples to isosystolic inequalities, Geom. Dedicata 57 (1995), 195--206.
  • —, Systolic Geometry and Topology, with an appendix by J. Solomon, Math. Surveys Monogr. 137, Amer. Math. Soc., Providence, 2007.
  • —, Systolic inequalities and Massey products in simply-connected manifolds, Israel J. Math. 164 (2008), 381--395.
  • M. G. Katz and C. Lescop, ``Filling area conjecture, optimal systolic inequalities, and the fiber class in abelian covers'' in Geometry, Spectral Theory, Groups, and Dynamics (Haifa, Israel, 2003--2004.), Contemp. Math. 387, Amer. Math. Soc., Providence, 2005, 181--200.
  • M. G. Katz, M. Schaps, and U. Vishne, Logarithmic growth of systole of arithmetic Riemann surfaces along congruence subgroups, J. Differential Geom. 76 (2007), 399--422.
  • M. G. Katz and S. Shnider, Cayley $4$-form comass and triality isomorphisms, preprint,\arxiv0801.0283v1[math.DG]
  • V. Y. Kraines, Topology of quaternionic manifolds, Trans. Amer. Math. Soc. 122 (1966), 357--367.
  • J.-H. Lee and N. C. Leung, Geometric structures on $G_2$ and $\Spin(7)$-manifolds, preprint,\arxivmath/0202045v2[math.DG]
  • P. M. Pu, Some inequalities in certain nonorientable Riemannian manifolds, Pacific J. Math. 2 (1952), 55--71.
  • Y. B. Rudyak and S. Sabourau, Systolic invariants of groups and $2$-complexes via Grushko decomposition, Ann. Inst. Fourier (Grenoble) 58 (2008), 777--800.
  • S. Sabourau, Asymptotic bounds for separating systoles on surfaces, Comment. Math. Helv. 83 (2008), 35--54.
  • S. L. Shatashvili and C. Vafa, Superstrings and manifolds of exceptional holonomy, Selecta Math. (N.S.) 1 (1995), 347--381.
  • H. Shiga, Rational homotopy type and self-maps, J. Math. Soc. Japan 31 (1979), 427--434.
  • è. B. Vinberg and A. G. èLašVili [Elashvili], A classification of the three-vectors of nine-dimensional space (in Russian), Trudy Sem. Vektor. Tenzor. Anal. 18 (1978), 197--233.
  • N. B. Wallach, Real Reductive Groups, I, Pure Appl. Math. 132, Academic Press, Boston, 1988.
  • G. W. Whitehead, Elements of Homotopy Theory, Grad. Texts in Math. 61, Springer, New York, 1978.
  • J. A. Wolf, Spaces of Constant Curvature, McGraw-Hill, New York, 1967.
  • A. H. Wright, ``Monotone mappings and degree one mappings between $PL$ manifolds'' in Geometric Topology (Park City, Utah, 1974), Lecture Notes in Math. 438, Springer, Berlin, 1975, 441--459.