Géométrie systolique et technique de régularisation
Séminaire de théorie spectrale et géométrie, Tome 31 (2012-2014), pp. 1-41.

L’objectif de ce texte est de présenter la notion de systole d’une variété riemannienne et de faire un survol de la géométrie systolique. On illustrera aussi une technique fondamentale, appelée technique de régularisation, qui est à la base de plusieurs résultats essentiels de géométrie systolique. Je détaillerai comment cette technique permet d’estimer les nombres de Betti d’une variété asphérique (d’après Gromov), et comment elle permet de relier l’entropie volumique à la systole et au volume systolique d’une variété riemannienne (d’après Sabourau).

DOI : 10.5802/tsg.292
Mots clés : Cycles géométriques, systole, volume systolique, espace d’Eilenberg-McLane, variété asphérique, nombres de Betti
Bulteau, Guillaume 1

1 Institut de Mathématiques et de Modélisation de Montpellier (I3M) UMR 5149 CNRS - Université Montpellier 2 Case courrier 051 34095 Montpellier cedex 5 (France)
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Bulteau, Guillaume. Géométrie systolique et technique de régularisation. Séminaire de théorie spectrale et géométrie, Tome 31 (2012-2014), pp. 1-41. doi : 10.5802/tsg.292. http://archive.numdam.org/articles/10.5802/tsg.292/

[1] Álvarez Paiva, J. C.; Balacheff, F. Contact geometry and isosystolic inequalities, Geom. Funct. Anal., Volume 24 (2014) no. 2, pp. 648-669 | DOI | MR | Zbl

[2] Babenko, Ivan K. Asymptotic invariants of smooth manifolds, Izv. Ross. Akad. Nauk Ser. Mat., Volume 56 (1992) no. 4, pp. 707-751 | DOI | MR | Zbl

[3] Babenko, Ivan K. Topologie des systoles unidimensionnelles, Enseign. Math. (2), Volume 52 (2006) no. 1-2, pp. 109-142 | MR | Zbl

[4] Babenko, Ivan K.; Balacheff, Florent Systolic volume of homology classes (2010) (http://arxiv.org/abs/1009.2835)

[5] Babenko, Ivan K.; Balacheff, Florent; Bulteau, Guillaume Systolic geometry and simplicial complexity for groups (2015) (http://arxiv.org/abs/1501.01173)

[6] Balacheff, Florent; Parlier, Hugo; Sabourau, Stéphane Short loop decompositions of surfaces and the geometry of Jacobians, Geom. Funct. Anal., Volume 22 (2012) no. 1, pp. 37-73 | DOI | MR | Zbl

[7] Bavard, C. Inégalité isosystolique pour la bouteille de Klein, Math. Ann., Volume 274 (1986) no. 3, pp. 439-441 | DOI | MR | Zbl

[8] Berger, Marcel À l’ombre de Loewner, Ann. Sci. École Norm. Sup. (4), Volume 5 (1972), pp. 241-260 | Numdam | MR | Zbl

[9] Berger, Marcel Du côté de chez Pu, Ann. Sci. École Norm. Sup. (4), Volume 5 (1972), pp. 1-44 | Numdam | MR | Zbl

[10] Berger, Marcel Une borne inférieure pour le volume d’une variété riemannienne en fonction du rayon d’injectivité, Ann. Inst. Fourier (Grenoble), Volume 30 (1980) no. 3, pp. 259-265 | Numdam | MR | Zbl

[11] Berger, Marcel Systoles et applications selon Gromov, Astérisque (1993) no. 216, pp. Exp. No. 771, 5, 279-310 (Séminaire Bourbaki, Vol. 1992/93) | Numdam | MR | Zbl

[12] Berger, Marcel A panoramic view of Riemannian geometry, Springer-Verlag, Berlin, 2003, pp. xxiv+824 | MR | Zbl

[13] Besson, G.; Courtois, G.; Gallot, S. Volume et entropie minimale des espaces localement symétriques, Invent. Math., Volume 103 (1991) no. 2, pp. 417-445 | DOI | MR | Zbl

[14] Brunnbauer, Michael Homological invariance for asymptotic invariants and systolic inequalities, Geom. Funct. Anal., Volume 18 (2008) no. 4, pp. 1087-1117 | DOI | MR | Zbl

[15] Bulteau, Guillaume Cycles géométriques réguliers (à paraître, Bull. SMF)

[16] Burago, Dmitri; Burago, Yuri; Ivanov, Sergei A course in metric geometry, Graduate Studies in Mathematics, 33, American Mathematical Society, Providence, RI, 2001, pp. xiv+415 | MR

[17] Burago, Yu. D.; Zalgaller, V. A. Geometric inequalities, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 285, Springer-Verlag, Berlin, 1988, pp. xiv+331 (Translated from the Russian by A. B. Sosinskiĭ, Springer Series in Soviet Mathematics) | DOI | MR | Zbl

[18] Buser, P.; Sarnak, P. On the period matrix of a Riemann surface of large genus, Invent. Math., Volume 117 (1994) no. 1, pp. 27-56 (With an appendix by J. H. Conway and N. J. A. Sloane) | DOI | MR | Zbl

[19] Croke, Christopher B. Some isoperimetric inequalities and eigenvalue estimates, Ann. Sci. École Norm. Sup. (4), Volume 13 (1980) no. 4, pp. 419-435 | Numdam | MR | Zbl

[20] Croke, Christopher B.; Katz, Mikhail Universal volume bounds in Riemannian manifolds, Surveys in differential geometry, Vol. VIII (Boston, MA, 2002) (Surv. Differ. Geom.), Volume 8, Int. Press, Somerville, MA, 2003, pp. 109-137 | DOI | MR | Zbl

[21] Dugundji, James Topology, Allyn and Bacon, Inc., Boston, Mass., 1966, pp. xvi+447 | MR | Zbl

[22] Gallot, Sylvestre; Hulin, Dominique; Lafontaine, Jacques Riemannian geometry, Universitext, Springer-Verlag, Berlin, 1990, pp. xiv+284 | MR | Zbl

[23] Gromov, Mikhael Structures métriques pour les variétés riemanniennes, Textes Mathématiques [Mathematical Texts], 1, CEDIC, Paris, 1981, pp. iv+152 (Edited by J. Lafontaine and P. Pansu) | MR

[24] Gromov, Mikhael Volume and bounded cohomology, Inst. Hautes Études Sci. Publ. Math. (1982) no. 56, p. 5-99 (1983) | Numdam | MR | Zbl

[25] Gromov, Mikhael Filling Riemannian manifolds, J. Differential Geom., Volume 18 (1983) no. 1, pp. 1-147 http://projecteuclid.org/getRecord?id=euclid.jdg/1214509283 | MR | Zbl

[26] Gromov, Mikhael Systoles and intersystolic inequalities, Actes de la Table Ronde de Géométrie Différentielle (Luminy, 1992) (Sémin. Congr.), Volume 1, Soc. Math. France, Paris, 1996, pp. 291-362 | MR | Zbl

[27] Gromov, Mikhael Metric structures for Riemannian and non-Riemannian spaces, Progress in Mathematics, 152, Birkhäuser Boston Inc., Boston, MA, 1999, pp. xx+585 Based on the 1981 French original [ MR0682063 (85e :53051)], With appendices by M. Katz, P. Pansu and S. Semmes, Translated from the French by Sean Michael Bates | MR | Zbl

[28] Guth, Larry Notes on Gromov’s systolic estimate, Geom. Dedicata, Volume 123 (2006), pp. 113-129 | DOI | MR | Zbl

[29] Guth, Larry Metaphors in systolic geometry, Proceedings of the International Congress of Mathematicians. Volume II, Hindustan Book Agency, New Delhi (2010), pp. 745-768 | MR | Zbl

[30] Hatcher, Allen Algebraic topology, Cambridge University Press, Cambridge, 2002, pp. xii+544 | MR | Zbl

[31] Hebda, James J. Some lower bounds for the area of surfaces, Invent. Math., Volume 65 (1981/82) no. 3, pp. 485-490 | DOI | MR | Zbl

[32] Katok, A. Entropy and closed geodesics, Ergodic Theory Dynam. Systems, Volume 2 (1982) no. 3-4, p. 339-365 (1983) | DOI | MR | Zbl

[33] Katok, Anatole; Hasselblatt, Boris Introduction to the modern theory of dynamical systems, Encyclopedia of Mathematics and its Applications, 54, Cambridge University Press, Cambridge, 1995, pp. xviii+802 (With a supplementary chapter by Katok and Leonardo Mendoza) | DOI | MR | Zbl

[34] Katz, Karin Usadi; Katz, Mikhail G.; Sabourau, Stéphane; Shnider, Steven; Weinberger, Shmuel Relative systoles of relative-essential 2-complexes, Algebr. Geom. Topol., Volume 11 (2011) no. 1, pp. 197-217 | DOI | MR | Zbl

[35] Katz, Mikhail G. Systolic geometry and topology, Mathematical Surveys and Monographs, 137, American Mathematical Society, Providence, RI, 2007, pp. xiv+222 (With an appendix by Jake P. Solomon) | DOI | MR | Zbl

[36] Katz, Mikhail G.; Sabourau, Stéphane Entropy of systolically extremal surfaces and asymptotic bounds, Ergodic Theory Dynam. Systems, Volume 25 (2005) no. 4, pp. 1209-1220 | DOI | MR | Zbl

[37] Kodani, Shigeru On two-dimensional isosystolic inequalities, Kodai Math. J., Volume 10 (1987) no. 3, pp. 314-327 | DOI | MR | Zbl

[38] Manning, Anthony Topological entropy for geodesic flows, Ann. of Math. (2), Volume 110 (1979) no. 3, pp. 567-573 | DOI | MR | Zbl

[39] Pu, P. M. Some inequalities in certain nonorientable Riemannian manifolds, Pacific J. Math., Volume 2 (1952), pp. 55-71 | MR | Zbl

[40] Reviron, Guillemette Rigidité topologique sous l’hypothèse “entropie majorée” et applications, Comment. Math. Helv., Volume 83 (2008) no. 4, pp. 815-846 | DOI | MR | Zbl

[41] Rudyak, Yuli B.; Sabourau, Stéphane Systolic invariants of groups and 2-complexes via Grushko decomposition, Ann. Inst. Fourier (Grenoble), Volume 58 (2008) no. 3, pp. 777-800 | Numdam | MR | Zbl

[42] Sabourau, Stéphane Systolic volume and minimal entropy of aspherical manifolds, J. Differential Geom., Volume 74 (2006) no. 1, pp. 155-176 http://projecteuclid.org/getRecord?id=euclid.jdg/1175266185 | MR | Zbl

[43] Sakai, Takashi A proof of the isosystolic inequality for the Klein bottle, Proc. Amer. Math. Soc., Volume 104 (1988) no. 2, pp. 589-590 | DOI | MR | Zbl

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