Systolic almost-rigidity modulo 2

[en] No power law systolic freedom is possible for the product of mod 2 systoles of dimension 1 and codimension 1. This means that any closed n-dimensional Riemannian manifold M of bounded local geometry obeys the following systolic inequality: the product of its mod 2 systoles of dimensions 1 and n-1 is bounded from above by c(n,\epsilon) Vol(M)^{1+\epsilon}, if finite (if H_1(M; Z/2) is non-trivial).

Disciplines :

Mathematics

Author, co-author :

Alpert, Hannah ; Auburn University, Auburn, USA

BALITSKIY, Alexey ; University of Luxembourg ; Institute for Advanced Study, Princeton, USA

Guth, Larry; Massachusetts Institute of Technology, Cambridge, USA

External co-authors :

yes

Language :

English

Title :

Systolic almost-rigidity modulo 2

Publication date :

25 June 2024

Journal title :

Journal of the European Mathematical Society

ISSN :

1435-9855

eISSN :

1435-9863

Publisher :

European Mathematical Society - EMS - Publishing House GmbH

Peer reviewed :

Peer Reviewed verified by ORBi

Additional URL :

https://ems.press/journals/jems/articles/14297929
https://arxiv.org/abs/2206.01968

Funders :

Horizon Europe
NSF - National Science Foundation

Available on ORBilu :

since 15 October 2025

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Bibliography

Avvakumov, S., Balitskiy, A., Hubard, A., Karasev, R.: Systolic inequalities for the number of vertices. J. Topol. Anal. (online, 2023)
Babenko, I., Katz, M.: Systolic freedom of orientable manifolds. Ann. Sci. École Norm. Sup. (4) 31, 787–809 (1998) Zbl 0944.53019 MR 1664222
Balacheff, F., Karam, S.: Length product of homologically independent loops for tori. J. Topol. Anal. 8, 497–500 (2016) Zbl 1342.53060 MR 3509570
Berger, M.: A panoramic view of Riemannian geometry. Springer, Berlin (2003) Zbl 1038.53002 MR 2002701
Boissonnat, J.-D., Dyer, R., Ghosh, A.: Delaunay triangulation of manifolds. Found. Comput. Math. 18, 399–431 (2018) Zbl 1395.57032 MR 3777784
Boissonnat, J.-D., Dyer, R., Ghosh, A., Wintraecken, M.: Local criteria for triangulating general manifolds. Discrete Comput. Geom. 69, 156–191 (2023) Zbl 1521.57023 MR 4527538
Bowditch, B. H.: Bilipschitz triangulations of riemannian manifolds. Preprint, http://homepages.warwick.ac.uk/~masgak/papers/triangulations.pdf (2020), visited on June 12, 2024
Cairns, S. S.: On the triangulation of regular loci. Ann. of Math. (2) 35, 579–587 (1934) Zbl 0012.03605 MR 1503181
Dyer, R., Vegter, G., Wintraecken, M.: Riemannian simplices and triangulations. Geom. Dedicata 179, 91–138 (2015) Zbl 1333.57036 MR 3424659
Freedman, M., Hastings, M.: Building manifolds from quantum codes. Geom. Funct. Anal. 31, 855–894 (2021) Zbl 1485.53115 MR 4317505
Freedman, M. H.: Z2-systolic-freedom. In: Proceedings of the Kirbyfest (Berkeley, CA, 1998), Geom. Topol. Monogr. 2, Geometry & Topology Publications, Coventry, 113–123 (1999) Zbl 1024.53029 MR 1734404
Freedman, M. H., Meyer, D. A., Luo, F.: Z2-systolic freedom and quantum codes. In: Mathematics of quantum computation, Comput. Math. Ser., Chapman & Hall/CRC, Boca Raton, FL, 287–320 (2002) Zbl 1075.81508 MR 2007952
Gromov, M.: Systoles and intersystolic inequalities. In: Actes de la Table Ronde de Géométrie Différentielle (Luminy, 1992), Sémin. Congr. 1, Société Mathématique de France, Paris, 291–362 (1996) MR 1427763
Guth, L.: Systolic inequalities and minimal hypersurfaces. Geom. Funct. Anal. 19, 1688–1692 (2010) Zbl 1185.53045 MR 2594618
Hastings, M. B., Haah, J., O’Donnell, R.: Fiber bundle codes: breaking the N1=2polylog.N/barrier for quantum LDPC codes. In: STOC ’21—Proceedings of the 53rd Annual ACM SIGACT Symposium on Theory of Computing, ACM, New York, 1276–1288 (2021) Zbl 07765248 MR 4398919
Katz, M. G., Suciu, A. I.: Volume of Riemannian manifolds, geometric inequalities, and homotopy theory. In: Tel Aviv Topology Conference: Rothenberg Festschrift (1998), Contemp. Math. 231, American Mathematical Society, Providence, RI, 113–136 (1999) Zbl 0967.53024 MR 1705579
Panteleev, P., Kalachev, G.: Quantum LDPC codes with almost linear minimum distance. IEEE Trans. Inform. Theory 68, 213–229 (2022) Zbl 1489.94162 MR 4395409
Papasoglu, P.: Uryson width and volume. Geom. Funct. Anal. 30, 574–587 (2020) Zbl 1453.53050 MR 4108616
Pittet, C.: Systoles on S1Sn. Differential Geom. Appl. 7, 139–142 (1997) Zbl 0878.53038 MR 1454720
Whitehead, J. H. C.: On C1-complexes. Ann. of Math. (2) 41, 809–824 (1940) Zbl 0025.09203 MR 0002545