Stable homology of surface diffeomorphism groups made discrete

Nariman, Sam¹

¹ Department of Mathematics, Northwestern University, Evanston, IL, United States

Geometry & topology, Tome 21 (2017) no. 5, pp. 3047-3092.

Résumé

We answer affirmatively a question posed by Morita on homological stability of surface diffeomorphisms made discrete. In particular, we prove that $C^{\infty }$–diffeomorphisms of surfaces as family of discrete groups exhibit homological stability. We show that the stable homology of $C^{\infty }$–diffeomorphisms of surfaces as discrete groups is the same as homology of certain infinite loop space related to Haefliger’s classifying space of foliations of codimension $2$. We use this infinite loop space to obtain new results about (non)triviality of characteristic classes of flat surface bundles and codimension-$2$ foliations.

Détail

DOI : 10.2140/gt.2017.21.3047

Classification : 58D05, 57R32, 55P35, 55R40, 57R19, 57R32, 57R50, 57R20
Keywords: Discrete diffeomorphisms, Haefliger classifying space, Homological stability, infinite loop space

Bibliographie

[1] T Akita, N Kawazumi, T Uemura, Periodic surface automorphisms and algebraic independence of Morita–Mumford classes, J. Pure Appl. Algebra 160 (2001) 1 | DOI

[2] R Bott, Lectures on characteristic classes and foliations, from: "Lectures on algebraic and differential topology" (editor S Gitler), Lecture Notes in Math. 279, Springer (1972) 1 | DOI

[3] J Bowden, The homology of surface diffeomorphism groups and a question of Morita, Proc. Amer. Math. Soc. 140 (2012) 2543 | DOI

[4] J Cheeger, J Simons, Differential characters and geometric invariants, from: "Geometry and topology" (editors J Alexander, J Harer), Lecture Notes in Math. 1167, Springer (1985) 50 | DOI

[5] S Galatius, Mod p homology of the stable mapping class group, Topology 43 (2004) 1105 | DOI

[6] S Galatius, I Madsen, U Tillmann, Divisibility of the stable Miller–Morita–Mumford classes, J. Amer. Math. Soc. 19 (2006) 759 | DOI

[7] S Galatius, O Randal-Williams, Monoids of moduli spaces of manifolds, Geom. Topol. 14 (2010) 1243 | DOI

[8] S Galatius, O Randal-Williams, Stable moduli spaces of high-dimensional manifolds, Acta Math. 212 (2014) 257 | DOI

[9] S Galatius, U Tillmann, I Madsen, M Weiss, The homotopy type of the cobordism category, Acta Math. 202 (2009) 195 | DOI

[10] A Haefliger, Homotopy and integrability, from: "Manifolds" (editor N H Kuiper), Lecture Notes in Math. 197, Springer (1971) 133 | DOI

[11] J Harer, The second homology group of the mapping class group of an orientable surface, Invent. Math. 72 (1983) 221 | DOI

[12] J L Harer, Stability of the homology of the mapping class groups of orientable surfaces, Ann. of Math. 121 (1985) 215 | DOI

[13] J Harer, The third homology group of the moduli space of curves, Duke Math. J. 63 (1991) 25 | DOI

[14] S Hurder, Foliation geometry/topology problem set, preprint (2003)

[15] U Koschorke, Singularities and bordism of q–plane fields and of foliations, Bull. Amer. Math. Soc. 80 (1974) 760 | DOI

[16] D Kotschick, S Morita, Signatures of foliated surface bundles and the symplectomorphism groups of surfaces, Topology 44 (2005) 131 | DOI

[17] D Kotschick, S Morita, Characteristic classes of foliated surface bundles with area-preserving holonomy, J. Differential Geom. 75 (2007) 273 | DOI

[18] H B Lawson Jr., The quantitative theory of foliations, 27, Amer. Math. Soc. (1977)

[19] I Madsen, U Tillmann, The stable mapping class group and Q(CP+∞), Invent. Math. 145 (2001) 509 | DOI

[20] I Madsen, M Weiss, The stable moduli space of Riemann surfaces : Mumford’s conjecture, Ann. of Math. 165 (2007) 843 | DOI

[21] J N Mather, Commutators of diffeomorphisms, Comment. Math. Helv. 49 (1974) 512 | DOI

[22] J Mather, On the homology of Haefliger’s classifying space, from: "Differential topology" (editor V Villani), CIME Summer Schools 73, Springer (2011) 71 | DOI

[23] J P May, K Ponto, More concise algebraic topology: localization, completion, and model categories, University of Chicago Press (2012)

[24] D Mcduff, Local homology of groups of volume-preserving diffeomorphisms, III, Ann. Sci. École Norm. Sup. 16 (1983) 529 | DOI

[25] J Milnor, The geometric realization of a semi-simplicial complex, Ann. of Math. 65 (1957) 357 | DOI

[26] J Milnor, Microbundles, I, Topology 3 (1964) 53 | DOI

[27] J Milnor, On the homology of Lie groups made discrete, Comment. Math. Helv. 58 (1983) 72 | DOI

[28] T Mizutani, S Morita, T Tsuboi, On the cobordism classes of codimension one foliations which are almost without holonomy, Topology 22 (1983) 325 | DOI

[29] S Morita, Discontinuous invariants of foliations, from: "Foliations" (editor I Tamura), Adv. Stud. Pure Math. 5, North-Holland (1985) 169

[30] S Morita, Characteristic classes of surface bundles, Invent. Math. 90 (1987) 551 | DOI

[31] S Morita, Geometry of characteristic classes, 199, Amer. Math. Soc. (2001)

[32] S Morita, Cohomological structure of the mapping class group and beyond, from: "Problems on mapping class groups and related topics" (editor B Farb), Proc. Sympos. Pure Math. 74, Amer. Math. Soc. (2006) 329 | DOI

[33] S Nariman, Homological stability and stable moduli of flat manifold bundles, preprint (2014)

[34] S Nariman, On the moduli space of flat symplectic surface bundles, preprint (2016)

[35] H V Pittie, Characteristic classes of foliations, 10, Pitman (1976)

[36] J Powell, Two theorems on the mapping class group of a surface, Proc. Amer. Math. Soc. 68 (1978) 347 | DOI

[37] O Randal-Williams, Resolutions of moduli spaces and homological stability, J. Eur. Math. Soc. 18 (2016) 1 | DOI

[38] O H Rasmussen, Continuous variation of foliations in codimension two, Topology 19 (1980) 335 | DOI

[39] Y B Rudyak, On Thom spectra, orientability, and cobordism, Springer (1998) | DOI

[40] F Sergeraert, BΓ [d’après John N Mather et William Thurston], from: "Séminaire Bourbaki (1977/78)", Lecture Notes in Math. 710, Springer (1979) 300

[41] S Smale, Diffeomorphisms of the 2–sphere, Proc. Amer. Math. Soc. 10 (1959) 621 | DOI

[42] P Teichner, Topological four-manifolds with finite fundamental group, Doktorarbeit, Johannes Gutenberg-Universität Mainz (1992)

[43] W Thurston, Foliations and groups of diffeomorphisms, Bull. Amer. Math. Soc. 80 (1974) 304 | DOI

[44] U Tillmann, On the homotopy of the stable mapping class group, Invent. Math. 130 (1997) 257 | DOI

[45] T Tsuboi, On the foliated products of class C1, Ann. of Math. 130 (1989) 227 | DOI

[46] T Tsuboi, Classifying spaces for groupoid structures, from: "Foliations, geometry, and topology" (editors N C Saldanha, L Conlon, R Langevin, T Tsuboi, P Walczak), Contemp. Math. 498, Amer. Math. Soc. (2009) 67 | DOI