Duke Mathematical Journal

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

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

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Abstract

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

Article information

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

Dates
First available in Project Euclid: 17 December 2008

Permanent link to this document
http://projecteuclid.org/euclid.dmj/1229530284

Digital Object Identifier
doi:10.1215/00127094-2008-061

Subjects
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

Citation

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.


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References

  • 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.