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Extending Gromov’s optimal systolic inequality

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Abstract

The existence of nontrivial cup products or Massey products in the cohomology of a manifold leads to inequalities of systolic type, but in general such inequalities are not optimal (tight). Gromov proved an optimal systolic inequality for complex projective space. We provide a natural extension of Gromov’s inequality to manifolds whose fundamental cohomology class is a cup product of 2-dimensional classes.

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Notes

  1. Gromov [12, 13].

  2. Katz and Lescop [18]; Katz [17].

  3. See [6, 7, 19].

  4. John [15, p. 203]; Milman and Schechtman [23, Section 3.3, p. 10].

  5. Speyer https://mathoverflow.net/a/449498/28128.

  6. Banaszczyk [2].

  7. Banaszczyk [3].

  8. Without the stabilisation (i.e., allowing denominators in cycles), one witnesses a widespread phenomenon of systolic freedom; see e.g., Babenko and Katz [1].

  9. Whitney [27]; Federer [10, Section 4.10, p. 380]; Gromov [13, Section 4.34, p. 261]; Pansu [24, Lemma 17].

  10. See also Goodwillie https://mathoverflow.net/a/449042/28128 The bound also results by representing the cup product by a suitable Pfaffian, and applying Roos [26, Lemma 2.1, p. 1788] (see there for some history; the result goes back to Banach).

  11. Gromov [11, item 4.37, p. 60].

  12. Gromov [13, item 4.37, p. 262].

  13. Bangert et al. [4, Proposition 1.4]).

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Acknowledgements

The authors are grateful to Péter Ernő Frenkel for bringing the article by Roos [26] to our attention, to Gennadiy Averkov for bringing the articles [22] and [16] to our attention, and to Oliver Knill, Emanuel Lazar, and Steve Shnider for reading several preliminary versions of the text.

Funding

Mikhail Katz is partially supported by the BSF Grant 2020124 and the ISF Grant 743/22.

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Authors and Affiliations

  1. Department of Mathematics, Brown University, Providence, USA

    Thomas G. Goodwillie

  2. Department of Mathematics and Statistics, Saint Louis University, 220 N. Grand Boulevard, St. Louis, MO, 63103, USA

    James J. Hebda

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

    Mikhail G. Katz

Authors
  1. Thomas G. Goodwillie
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  2. James J. Hebda
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  3. Mikhail G. Katz
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TG, JH, and MK contributed equally to this work.

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Correspondence to Mikhail G. Katz.

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Goodwillie, T.G., Hebda, J.J. & Katz, M.G. Extending Gromov’s optimal systolic inequality. J. Geom. 114, 23 (2023). https://doi.org/10.1007/s00022-023-00685-3

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