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Cohomological dimension, self-linking, and systolic geometry

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Abstract

Given a closed manifold M, we prove the upper bound of

$${1 \over 2}(\dim M + {\rm{cd}}({{\rm{\pi }}_1}M))$$

for the number of systolic factors in a curvature-free lower bound for the total volume of M, in the spirit of M. Gromov’s systolic inequalities. Here “cd” is the cohomological dimension. We apply this upper bound to show that, in the case of a 4-manifold, the Lusternik-Schnirelmann category is an upper bound for the systolic category. Furthermore, we prove a systolic inequality on a manifold M with b 1(M) = 2 in the presence of a nontrivial self-linking class of a typical fiber of its Abel-Jacobi map to the 2-torus.

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References

  1. L. Ambrosio and M. Katz, Flat currents modulo p in metric spaces and filling radius inequalities, Commentarii Mathematici Helvetici. A Journal of the Swiss Mathematical Society, to appear.

  2. I. Babenko, Asymptotic invariants of smooth manifolds, Russian Academy of Sciences. Izvestiya Mathematics 41 (1993), 1–38.

    Article  MathSciNet  Google Scholar 

  3. I. Babenko, Loewner’s conjecture, Besicovich’s example, and relative systolic geometry, Matematichskiĭ Sbornik 193 (2002), 3–16.

    MathSciNet  Google Scholar 

  4. I. Babenko, Topologie des systoles unidimensionelles, L’Enseignement Mathématique (2) 52 (2006), 109–142.

    MathSciNet  MATH  Google Scholar 

  5. V. Bangert, C. Croke, S. Ivanov and M. Katz, Boundary case of equality in optimal Loewner-type inequalities, Transactions of the American Mathematical Society 359 (2007), 1–17. See arXiv:math.DG/0406008

    Article  MathSciNet  MATH  Google Scholar 

  6. V. Bangert and M. Katz, Stable systolic inequalities and cohomology products, Communications on Pure and Applied Mathematics 56 (2003), 979–997. Available at the site arXiv:math.DG/0204181

    Article  MathSciNet  MATH  Google Scholar 

  7. V. Bangert and M. Katz, An optimal Loewner-type systolic inequality and harmonic one-forms of constant norm, Communications in Analysis and Geometry 12 (2004), 703–732. See arXiv:math.DG/0304494

    MathSciNet  MATH  Google Scholar 

  8. V. Bangert, M. Katz, S. Shnider and S. Weinberger, E 7, Wirtinger inequalities, Cayley 4-form, and homotopy, Duke Mathematical Journal 146 (2009), 35–70. See arXiv:math.DG/0608006

    Article  MathSciNet  MATH  Google Scholar 

  9. M. Berger, Du côté de chez Pu, Annales Scientifiques de l’École Normale Supérieure. Quatriéme Série 5 (1972), 1–44.

    MATH  Google Scholar 

  10. M. Berger, A l’ombre de Loewner Annales Scientifiques de l’École Normale Supérieure. Quatriéme Série 5 (1972), 241–260.

    MATH  Google Scholar 

  11. M. Berger, Systoles et applications selon Gromov, Séminaire N. Bourbaki, exposé 771, Astérisque 216 (1993), 279–310.

    Google Scholar 

  12. M. Berger, A Panoramic View of Riemannian Geometry, Springer-Verlag, Berlin, 2003.

    Book  MATH  Google Scholar 

  13. M. Berger, What is... a Systole? Notices of the American Mathematical Society 55 (2008), 374–376.

    MathSciNet  MATH  Google Scholar 

  14. K. Brown, Cohomology of Groups, Graduate Texts in Mathematics 87, Springer, New York, 1994.

    Google Scholar 

  15. M. Brunnbauer, Homological invariance for asymptotic invariants and systolic inequalities, Geometric and Functional Analysis (GAFA) 18 (2008), 1087–1117. See arXiv:math.GT/0702789.

    Article  MathSciNet  MATH  Google Scholar 

  16. M. Brunnbauer, Filling inequalities do not depend on topology, Journal für die Reine und Angewandte Mathematik 624 (2008), 217–231. See arXiv:0706.2790.

    Article  MathSciNet  MATH  Google Scholar 

  17. M. Brunnbauer, On manifolds satisfying stable systolic inequalities, Mathematische Annalen 342 (2008), 951–968. See arXiv:0708.2589.

    Article  MathSciNet  MATH  Google Scholar 

  18. T. Cochran and J. Masters, The growth rate of the first Betti number in abelian covers of 3-manifolds, Mathematical Proceedings of the Cambridge Philosophical Society 141 (2006), 465–476. See arXiv:math.GT/0508294.

    Article  MathSciNet  MATH  Google Scholar 

  19. O. Cornea, G. Lupton, J. Oprea and D. Tanré, Lusternik-Schnirelmann category, Mathematical Surveys and Monographs 103, American Mathematical Society, Providence, RI, 2003.

    Google Scholar 

  20. A. Costa and M. Farber, Motion planning in spaces with small fundamental group, Communications in Contemporary Mathematics 12 (2010), 107–119.

    Article  MathSciNet  MATH  Google Scholar 

  21. A. Dranishnikov, On the Lusternik-Schnirelmann category of spaces with 2 dimensional fundamental group, Proceedings of the American Mathematical Society 137 (2009), 1489–1497. See arXiv:0709.4018.

    Article  MathSciNet  MATH  Google Scholar 

  22. A. Dranishnikov, The Lusternik-Schnirelmann category and the fundamental group, Algebraic & Geometric Topology 10 (2010), 917–924.

    Article  MathSciNet  MATH  Google Scholar 

  23. A. Dranishnikov, M. Katz and Y. Rudyak, Small values of the Lusternik-Schnirelmann category for manifolds, Geometry and Topology 12 (2008), 1711–1727. See arXiv:0805.1527

    Article  MathSciNet  MATH  Google Scholar 

  24. A. Dranishnikov and Yu. Rudyak, On the Berstein-Svarc Theorem in dimension 2, Mathematical Proceedings of the Cambridge Philosophical Society 146 (2009), 407–413.

    Article  MathSciNet  MATH  Google Scholar 

  25. A. Dranishnikov and Y. Rudyak, Stable systolic category of manifolds and the cup-length, Journal of Fixed Point Theory and Applications 6 (2009), 165–177.

    Article  MathSciNet  MATH  Google Scholar 

  26. S. Eilenberg and T. Ganea, On the Lusternik-Schnirelmann category of abstract groups, Annals of Mathematics. Second Series 65 (1957), 517–518.

    Article  MathSciNet  Google Scholar 

  27. C. Elmir and J. Lafontaine, Sur la géométrie systolique des variétés de Bieberbach, Geometrie Dedicata 136 (2008), 95–110.

    Article  MathSciNet  Google Scholar 

  28. H. Federer, Real flat chains, cochain and variational problems Indiana University Mathematics Journal 24 (1974), 351–407.

    Article  MathSciNet  MATH  Google Scholar 

  29. M. Gromov, Filling Riemannian manifolds, Journal of Differential Geometry 18 (1983), 1–147.

    MathSciNet  MATH  Google Scholar 

  30. M. Gromov, Systoles and intersystolic inequalities, in Actes de la Table Ronde de Géométrie Différentielle (Luminy, 1992), Séminaires et Congrés., 1, Soc. Math. France, Paris, 1996, pp, 291–362. www.emis.de/journals/SC/1996/1/ps/smf_sem-cong_1_291-362.ps.gz

    Google Scholar 

  31. M. Gromov, Metric structures for Riemannian and non-Riemannian spaces, Progress in Mathematics 152, Birkhäuser, Boston, 1999.

  32. M. Gromov, Metric Structures for Riemannian and Non-Riemannian Spaces, Based on the 1981 French original, with appendices by M. Katz, P. Pansu and S. Semmes, translated from the French by Sean Michael Bates. Reprint of the 2001 English edition, Modern Birkhäuser Classics, Birkhäuser Boston, Inc., Boston, MA, 2007.

    MATH  Google Scholar 

  33. L. Guth, Systolic inequalities and minimal hypersurfaces, Geometric and Functional Analysis 19 (2010), 1688–1692.

    Article  MathSciNet  MATH  Google Scholar 

  34. C. Horowitz, K. U. Katz and M. Katz, Loewner’s torus inequality with isosystolic defect, Journal of Geometric Analysis 19 (2009), 796–808. See arXiv:0803.0690.

    Article  MathSciNet  MATH  Google Scholar 

  35. S. Ivanov and M. Katz, Generalized degree and optimal Loewner-type inequalities, Israel Journal of Mathematics 141 (2004), 221–233. Available at the site arXiv:math.DG/0405019.

    Article  MathSciNet  MATH  Google Scholar 

  36. K. U. Katz and M. Katz, Hyperellipticity and Klein bottle companionship in systolic geometry, See arXiv:0811.1717

  37. K. U. Katz and M. Katz, Bi-Lipschitz approximation by finite-dimensional imbeddings 150 (2011), 131–136.

    Google Scholar 

  38. K. U. Katz, M. Katz, S. Sabourau, S. Shnider and Sh. Weinberger, Relative systoles of relative-essential 2-complexes, Algebraic and geometric topology 11 (2011), 197–217.

    Article  MathSciNet  MATH  Google Scholar 

  39. M. Katz, Systolic geometry and topology, with an appendix by Jake P. Solomon. Mathematical Surveys and Monographs, 137, American Mathematical Society, Providence, RI, 2007.

    MATH  Google Scholar 

  40. M. Katz, Systolic inequalities and Massey products in simply-connected manifolds, Israel Journal of Mathematics 164 (2008), 381–395. Available at the site arXiv:math.DG/0604012.

    Article  MathSciNet  MATH  Google Scholar 

  41. M. Katz and C. Lescop, Filling area conjecture, optimal systolic inequalities, and the fiber class in abelian covers, in Geometry, Spectral Theory, Groups, and Dynamics, Contemporary Mathematics 387, American Mathematical Society, Providence, RI, 2005, pp. 181–200. See arXiv:math.DG/0412011

    Google Scholar 

  42. M. Katz and Y. Rudyak, Lusternik-Schnirelmann category and systolic category of low dimensional manifolds, Communications on Pure and Applied Mathematics 59 (2006), 1433–1456. arXiv:math.DG/0410456.

    Article  MathSciNet  MATH  Google Scholar 

  43. M. Katz and Y. Rudyak, Bounding volume by systoles of 3-manifolds, Journal of the London Mathematical Society 78 (2008), 407–417. See arXiv:math.DG/0504008.

    Article  MathSciNet  MATH  Google Scholar 

  44. M. Katz and S. Shnider, Cayley 4-form comass and triality isomorphisms, Israel Journal of Mathematics 178 (2010), 187–208. See arXiv:0801.0283.

    Article  MathSciNet  MATH  Google Scholar 

  45. C. Lescop, Global surgery formula for the Casson-Walker invariant, Annals of Mathematics Studies 140, Princeton University Press, Princeton 1996.

    Google Scholar 

  46. A. Lichnerowicz, Applications harmoniques dans un tore, Comptes Rendus Hebdomadaires des Séances de l’Académie des Sciences. Séries A et B 269 (1969), 912–916.

    MathSciNet  MATH  Google Scholar 

  47. T. Matumoto and A. Katanaga, On 4-dimensional closed manifolds with free fundamental groups, Hiroshima Mathematical Journal 25 (1995), 367–370.

    MathSciNet  MATH  Google Scholar 

  48. P. M. Pu, Some inequalities in certain nonorientable Riemannian manifolds, Pacific Journal of Mathematics 2 (1952), 55–71.

    MathSciNet  MATH  Google Scholar 

  49. Yu. Rudyak, Category weight: new ideas concerning Lusternik-Schnirelmann category, in Homotopy and Geometry (Warsaw, 1997), Banach Center Publications, 45, Polish Acad. Sci., Warsaw, 1998, pp. 47–61.

    Google Scholar 

  50. Yu. Rudyak, On category weight and its applications, Topology 38 (1999), 37–55.

    Article  MathSciNet  MATH  Google Scholar 

  51. Y. Rudyak and S. Sabourau, Systolic invariants of groups and 2-complexes via Grushko decomposition, Université de Grenoble. Annales de l’Institut Fourier 58 (2008), 777–800. See arXiv:math.DG/0609379.

  52. S. Sabourau, Asymptotic bounds for separating systoles on surfaces, Commentarii Mathematici Helvetici 83 (2008), 35–54.

    Article  MathSciNet  MATH  Google Scholar 

  53. J. Strom, Lusternik-Schnirelmann category of spaces with free fundamental group, Algebraic & Geometric Topology 7 (2007), 1805–1808.

    Article  MathSciNet  MATH  Google Scholar 

  54. S. Wenger, Isoperimetric inequalities of Euclidean type in metric spaces, Geometric and Functional Analysis 15 (2005), 534–554.

    Article  MathSciNet  MATH  Google Scholar 

  55. B. White, The deformation theorem for flat chains, Acta Mathematica 183 (1999), 255–271.

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, University of Florida, 358 Little Hall, Gainesville, FL, 32611-8105, USA

    Alexander N. Dranishnikov & Yuli B. Rudyak

  2. Department of Mathematics, Bar Ilan University, Ramat Gan, 52900, Israel

    Mikhail G. Katz

Authors
  1. Alexander N. Dranishnikov
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  2. Mikhail G. Katz
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  3. Yuli B. Rudyak
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Correspondence to Alexander N. Dranishnikov.

Additional information

Supported by the NSF, grant DMS-0604494.

Supported by the Israel Science Foundation (grants 84/03 and 1294/06) and the BSF (grant 2006393).

Supported by the NSF, grant 0406311.

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Dranishnikov, A.N., Katz, M.G. & Rudyak, Y.B. Cohomological dimension, self-linking, and systolic geometry. Isr. J. Math. 184, 437–453 (2011). https://doi.org/10.1007/s11856-011-0075-8

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  • DOI: https://doi.org/10.1007/s11856-011-0075-8

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