Geometry & Topology

Stable homology of surface diffeomorphism groups made discrete

Sam Nariman

Full-text: Access denied (no subscription detected)

However, an active subscription may be available with MSP at msp.org/gt.

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

We answer affirmatively a question posed by Morita on homological stability of surface diffeomorphisms made discrete. In particular, we prove that C–diffeomorphisms of surfaces as family of discrete groups exhibit homological stability. We show that the stable homology of C–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.

Article information

Source
Geom. Topol., Volume 21, Number 5 (2017), 3047-3092.

Dates
Received: 3 March 2016
Revised: 17 October 2016
Accepted: 6 December 2016
First available in Project Euclid: 16 November 2017

Permanent link to this document
https://projecteuclid.org/euclid.gt/1510859281

Digital Object Identifier
doi:10.2140/gt.2017.21.3047

Mathematical Reviews number (MathSciNet)
MR3687114

Zentralblatt MATH identifier
1377.58004

Subjects
Primary: 58D05: Groups of diffeomorphisms and homeomorphisms as manifolds [See also 22E65, 57S05] 57R32: Classifying spaces for foliations; Gelfand-Fuks cohomology [See also 58H10] 55P35: Loop spaces 55R40: Homology of classifying spaces, characteristic classes [See also 57Txx, 57R20] 57R19: Algebraic topology on manifolds 57R32: Classifying spaces for foliations; Gelfand-Fuks cohomology [See also 58H10] 57R50: Diffeomorphisms
Secondary: 57R20: Characteristic classes and numbers

Keywords
Discrete diffeomorphisms Haefliger classifying space Homological stability infinite loop space

Citation

Nariman, Sam. Stable homology of surface diffeomorphism groups made discrete. Geom. Topol. 21 (2017), no. 5, 3047--3092. doi:10.2140/gt.2017.21.3047. https://projecteuclid.org/euclid.gt/1510859281


Export citation

References

  • T Akita, N Kawazumi, T Uemura, Periodic surface automorphisms and algebraic independence of Morita–Mumford classes, J. Pure Appl. Algebra 160 (2001) 1–11
  • R Bott, Lectures on characteristic classes and foliations, from “Lectures on algebraic and differential topology” (S Gitler, editor), Lecture Notes in Math. 279, Springer (1972) 1–94
  • J Bowden, The homology of surface diffeomorphism groups and a question of Morita, Proc. Amer. Math. Soc. 140 (2012) 2543–2549
  • J Cheeger, J Simons, Differential characters and geometric invariants, from “Geometry and topology” (J Alexander, J Harer, editors), Lecture Notes in Math. 1167, Springer (1985) 50–80
  • S Galatius, Mod $p$ homology of the stable mapping class group, Topology 43 (2004) 1105–1132
  • S Galatius, I Madsen, U Tillmann, Divisibility of the stable Miller–Morita–Mumford classes, J. Amer. Math. Soc. 19 (2006) 759–779
  • S Galatius, O Randal-Williams, Monoids of moduli spaces of manifolds, Geom. Topol. 14 (2010) 1243–1302
  • S Galatius, O Randal-Williams, Stable moduli spaces of high-dimensional manifolds, Acta Math. 212 (2014) 257–377
  • S Galatius, U Tillmann, I Madsen, M Weiss, The homotopy type of the cobordism category, Acta Math. 202 (2009) 195–239
  • A Haefliger, Homotopy and integrability, from “Manifolds” (N H Kuiper, editor), Lecture Notes in Math. 197, Springer (1971) 133–163
  • J Harer, The second homology group of the mapping class group of an orientable surface, Invent. Math. 72 (1983) 221–239
  • J L Harer, Stability of the homology of the mapping class groups of orientable surfaces, Ann. of Math. 121 (1985) 215–249
  • J Harer, The third homology group of the moduli space of curves, Duke Math. J. 63 (1991) 25–55
  • S Hurder, Foliation geometry/topology problem set, preprint (2003) \setbox0\makeatletter\@url http://homepages.math.uic.edu/~hurder/papers/58manuscript.pdf {\unhbox0
  • U Koschorke, Singularities and bordism of $q$–plane fields and of foliations, Bull. Amer. Math. Soc. 80 (1974) 760–765
  • D Kotschick, S Morita, Signatures of foliated surface bundles and the symplectomorphism groups of surfaces, Topology 44 (2005) 131–149
  • D Kotschick, S Morita, Characteristic classes of foliated surface bundles with area-preserving holonomy, J. Differential Geom. 75 (2007) 273–302
  • H B Lawson, Jr, The quantitative theory of foliations, Conference Board of the Mathematical Sciences Regional Conference Series in Mathematics 27, Amer. Math. Soc., Providence, RI (1977)
  • I Madsen, U Tillmann, The stable mapping class group and $Q(\mathbb C \mathrm{P}^\infty_+)$, Invent. Math. 145 (2001) 509–544
  • I Madsen, M Weiss, The stable moduli space of Riemann surfaces: Mumford's conjecture, Ann. of Math. 165 (2007) 843–941
  • J N Mather, Commutators of diffeomorphisms, Comment. Math. Helv. 49 (1974) 512–528
  • J N Mather, On the homology of Haefliger's classifying space, from “Differential topology” (V Villani, editor), CIME Summer Schools 73, Springer (2011) 71–116
  • J P May, K Ponto, More concise algebraic topology: localization, completion, and model categories, University of Chicago Press (2012)
  • D McDuff, Local homology of groups of volume-preserving diffeomorphisms, III, Ann. Sci. École Norm. Sup. 16 (1983) 529–540
  • J Milnor, The geometric realization of a semi-simplicial complex, Ann. of Math. 65 (1957) 357–362
  • J Milnor, Microbundles, I, Topology 3 (1964) 53–80
  • J Milnor, On the homology of Lie groups made discrete, Comment. Math. Helv. 58 (1983) 72–85
  • T Mizutani, S Morita, T Tsuboi, On the cobordism classes of codimension one foliations which are almost without holonomy, Topology 22 (1983) 325–343
  • S Morita, Discontinuous invariants of foliations, from “Foliations” (I Tamura, editor), Adv. Stud. Pure Math. 5, North-Holland, Amsterdam (1985) 169–193
  • S Morita, Characteristic classes of surface bundles, Invent. Math. 90 (1987) 551–577
  • S Morita, Geometry of characteristic classes, Translations of Mathematical Monographs 199, Amer. Math. Soc., Providence, RI (2001)
  • S Morita, Cohomological structure of the mapping class group and beyond, from “Problems on mapping class groups and related topics” (B Farb, editor), Proc. Sympos. Pure Math. 74, Amer. Math. Soc. (2006) 329–354
  • S Nariman, Homological stability and stable moduli of flat manifold bundles, preprint (2014)
  • S Nariman, On the moduli space of flat symplectic surface bundles, preprint (2016)
  • H V Pittie, Characteristic classes of foliations, Research Notes in Mathematics 10, Pitman, London (1976)
  • J Powell, Two theorems on the mapping class group of a surface, Proc. Amer. Math. Soc. 68 (1978) 347–350
  • O Randal-Williams, Resolutions of moduli spaces and homological stability, J. Eur. Math. Soc. 18 (2016) 1–81
  • O H Rasmussen, Continuous variation of foliations in codimension two, Topology 19 (1980) 335–349
  • Y B Rudyak, On Thom spectra, orientability, and cobordism, Springer (1998)
  • F Sergeraert, $B\Gamma $ [d'après John N Mather et William Thurston], from “Séminaire Bourbaki (1977/78)”, Lecture Notes in Math. 710, Springer (1979) exposé 524, 300–315
  • S Smale, Diffeomorphisms of the $2$–sphere, Proc. Amer. Math. Soc. 10 (1959) 621–626
  • P Teichner, Topological four-manifolds with finite fundamental group, Doktorarbeit, Johannes Gutenberg-Universität Mainz (1992) \setbox0\makeatletter\@url http://people.mpim-bonn.mpg.de/teichner/Papers/phd.pdf {\unhbox0
  • W Thurston, Foliations and groups of diffeomorphisms, Bull. Amer. Math. Soc. 80 (1974) 304–307
  • U Tillmann, On the homotopy of the stable mapping class group, Invent. Math. 130 (1997) 257–275
  • T Tsuboi, On the foliated products of class $C^1$, Ann. of Math. 130 (1989) 227–271
  • T Tsuboi, Classifying spaces for groupoid structures, from “Foliations, geometry, and topology” (N C Saldanha, L Conlon, R Langevin, T Tsuboi, P Walczak, editors), Contemp. Math. 498, Amer. Math. Soc., Providence, RI (2009) 67–81