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 : https://doi.org/10.5802/tsg.292
Mots clés : Cycles géométriques, systole, volume systolique, espace d’Eilenberg-McLane, variété asphérique, nombres de Betti 
@article{TSG_2012-2014__31__1_0,
     author = {Bulteau, Guillaume},
     title = {G\'eom\'etrie systolique et technique de r\'egularisation},
     journal = {S\'eminaire de th\'eorie spectrale et g\'eom\'etrie},
     pages = {1--41},
     publisher = {Institut Fourier},
     address = {Grenoble},
     volume = {31},
     year = {2012-2014},
     doi = {10.5802/tsg.292},
     language = {fr},
     url = {http://www.numdam.org/articles/10.5802/tsg.292/}
}
TY  - JOUR
AU  - Bulteau, Guillaume
TI  - Géométrie systolique et technique de régularisation
JO  - Séminaire de théorie spectrale et géométrie
PY  - 2012-2014
DA  - 2012-2014///
SP  - 1
EP  - 41
VL  - 31
PB  - Institut Fourier
PP  - Grenoble
UR  - http://www.numdam.org/articles/10.5802/tsg.292/
UR  - https://doi.org/10.5802/tsg.292
DO  - 10.5802/tsg.292
LA  - fr
ID  - TSG_2012-2014__31__1_0
ER  - 
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://www.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 | Article | MR 3192037 | Zbl 1292.53050

[2] Babenko, Ivan K. Asymptotic invariants of smooth manifolds, Izv. Ross. Akad. Nauk Ser. Mat., Volume 56 (1992) no. 4, pp. 707-751 | Article | MR 1208148 | Zbl 0812.57022

[3] Babenko, Ivan K. Topologie des systoles unidimensionnelles, Enseign. Math. (2), Volume 52 (2006) no. 1-2, pp. 109-142 | MR 2255530 | Zbl 1236.53037

[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 | Article | MR 2899682 | Zbl 1254.30057

[7] Bavard, C. Inégalité isosystolique pour la bouteille de Klein, Math. Ann., Volume 274 (1986) no. 3, pp. 439-441 | Article | MR 842624 | Zbl 0578.53032

[8] Berger, Marcel À l’ombre de Loewner, Ann. Sci. École Norm. Sup. (4), Volume 5 (1972), pp. 241-260 | Numdam | MR 309009 | Zbl 0237.53035

[9] Berger, Marcel Du côté de chez Pu, Ann. Sci. École Norm. Sup. (4), Volume 5 (1972), pp. 1-44 | Numdam | MR 309008 | Zbl 0227.52005

[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 597027 | Zbl 0421.53028

[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 1246401 | Zbl 0789.53040

[12] Berger, Marcel A panoramic view of Riemannian geometry, Springer-Verlag, Berlin, 2003, pp. xxiv+824 | MR 2002701 | Zbl 1038.53002

[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 | Article | MR 1085114 | Zbl 0723.53029

[14] Brunnbauer, Michael Homological invariance for asymptotic invariants and systolic inequalities, Geom. Funct. Anal., Volume 18 (2008) no. 4, pp. 1087-1117 | Article | MR 2465685 | Zbl 1160.53021

[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 1835418

[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) | Article | MR 936419 | Zbl 0633.53002

[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) | Article | MR 1269424 | Zbl 0814.14033

[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 608287 | Zbl 0465.53032

[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 | Article | MR 2039987 | Zbl 1051.53026

[21] Dugundji, James Topology, Allyn and Bacon, Inc., Boston, Mass., 1966, pp. xvi+447 | MR 193606 | Zbl 0397.54003

[22] Gallot, Sylvestre; Hulin, Dominique; Lafontaine, Jacques Riemannian geometry, Universitext, Springer-Verlag, Berlin, 1990, pp. xiv+284 | MR 1083149 | Zbl 1068.53001

[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 682063

[24] Gromov, Mikhael Volume and bounded cohomology, Inst. Hautes Études Sci. Publ. Math. (1982) no. 56, p. 5-99 (1983) | Numdam | MR 686042 | Zbl 0516.53046

[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 697984 | Zbl 0515.53037

[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 1427763 | Zbl 0877.53002

[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 1699320 | Zbl 0953.53002

[28] Guth, Larry Notes on Gromov’s systolic estimate, Geom. Dedicata, Volume 123 (2006), pp. 113-129 | Article | MR 2299729 | Zbl 1125.53028

[29] Guth, Larry Metaphors in systolic geometry, Proceedings of the International Congress of Mathematicians. Volume II (2010), pp. 745-768 | MR 2827817 | Zbl 1247.53044

[30] Hatcher, Allen Algebraic topology, Cambridge University Press, Cambridge, 2002, pp. xii+544 | MR 1867354 | Zbl 1044.55001

[31] Hebda, James J. Some lower bounds for the area of surfaces, Invent. Math., Volume 65 (1981/82) no. 3, pp. 485-490 | Article | MR 643566 | Zbl 0482.53028

[32] Katok, A. Entropy and closed geodesics, Ergodic Theory Dynam. Systems, Volume 2 (1982) no. 3-4, p. 339-365 (1983) | Article | MR 721728 | Zbl 0525.58027

[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) | Article | MR 1326374 | Zbl 0878.58020

[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 | Article | MR 2764040 | Zbl 1228.53056

[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) | Article | MR 2292367 | Zbl 1149.53003

[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 | Article | MR 2158402 | Zbl 1097.53027

[37] Kodani, Shigeru On two-dimensional isosystolic inequalities, Kodai Math. J., Volume 10 (1987) no. 3, pp. 314-327 | Article | MR 929991 | Zbl 0642.53042

[38] Manning, Anthony Topological entropy for geodesic flows, Ann. of Math. (2), Volume 110 (1979) no. 3, pp. 567-573 | Article | MR 554385 | Zbl 0426.58016

[39] Pu, P. M. Some inequalities in certain nonorientable Riemannian manifolds, Pacific J. Math., Volume 2 (1952), pp. 55-71 | MR 48886 | Zbl 0046.39902

[40] Reviron, Guillemette Rigidité topologique sous l’hypothèse “entropie majorée” et applications, Comment. Math. Helv., Volume 83 (2008) no. 4, pp. 815-846 | Article | MR 2442964 | Zbl 1153.53027

[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 2427510 | Zbl 1142.53035

[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 2260931 | Zbl 1112.53030

[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 | Article | MR 962833 | Zbl 0692.53019

Cité par Sources :