Géométrie systolique et technique de régularisation
Guillaume Bulteau1
1 Institut de Mathématiques et de Modélisation de Montpellier (I3M) UMR 5149 CNRS - Université Montpellier 2 Case courrier 051 34095 Montpellier cedex 5 (France)
Séminaire de théorie spectrale et géométrie, Volume 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
Keywords: Cycles géométriques, systole, volume systolique, espace d’Eilenberg-McLane, variété asphérique, nombres de Betti
Guillaume Bulteau 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) 
@article{TSG_2012-2014__31__1_0,
     author = {Guillaume Bulteau},
     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 = {https://proceedings.centre-mersenne.org/articles/10.5802/tsg.292/}
}
TY  - JOUR
AU  - Guillaume Bulteau
TI  - Géométrie systolique et technique de régularisation
JO  - Séminaire de théorie spectrale et géométrie
PY  - 2012-2014
SP  - 1
EP  - 41
VL  - 31
PB  - Institut Fourier
PP  - Grenoble
UR  - https://proceedings.centre-mersenne.org/articles/10.5802/tsg.292/
DO  - 10.5802/tsg.292
LA  - fr
ID  - TSG_2012-2014__31__1_0
ER  - 
%0 Journal Article
%A Guillaume Bulteau
%T Géométrie systolique et technique de régularisation
%J Séminaire de théorie spectrale et géométrie
%D 2012-2014
%P 1-41
%V 31
%I Institut Fourier
%C Grenoble
%U https://proceedings.centre-mersenne.org/articles/10.5802/tsg.292/
%R 10.5802/tsg.292
%G fr
%F TSG_2012-2014__31__1_0
Guillaume Bulteau. Géométrie systolique et technique de régularisation. Séminaire de théorie spectrale et géométrie, Volume 31 (2012-2014), pp. 1-41. doi : 10.5802/tsg.292. https://proceedings.centre-mersenne.org/articles/10.5802/tsg.292/

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

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

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

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

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

[6] Florent Balacheff; Hugo Parlier; Stéphane Sabourau 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] C. Bavard Inégalité isosystolique pour la bouteille de Klein, Math. Ann., Volume 274 (1986) no. 3, pp. 439-441 | DOI | MR | Zbl

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

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

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

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

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

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

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

[17] Yu. D. Burago; V. A. Zalgaller 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] P. Buser; P. Sarnak 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] Christopher B. Croke Some isoperimetric inequalities and eigenvalue estimates, Ann. Sci. École Norm. Sup. (4), Volume 13 (1980) no. 4, pp. 419-435 | Numdam | MR | Zbl

[20] Christopher B. Croke; Mikhail Katz 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] James Dugundji Topology, Allyn and Bacon, Inc., Boston, Mass., 1966, pp. xvi+447 | MR | Zbl

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

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

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

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

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

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

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

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

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

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

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

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

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

[41] Yuli B. Rudyak; Stéphane Sabourau 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] Stéphane Sabourau 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] Takashi Sakai 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

Cited by Sources: