Geometry & Topology

Relations among characteristic classes of manifold bundles

Ilya Grigoriev

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Abstract

We study relations among characteristic classes of smooth manifold bundles with highly connected fibers. For bundles with fiber the connected sum of g copies of a product of spheres Sd × Sd, where d is odd, we find numerous algebraic relations among so-called “generalized Miller–Morita–Mumford classes”. For all g > 1, we show that these infinitely many classes are algebraically generated by a finite subset.

Our results contrast with the fact that there are no algebraic relations among these classes in a range of cohomological degrees that grows linearly with g, according to recent homological stability results. In the case of surface bundles (d = 1), our approach recovers some previously known results about the structure of the classical “tautological ring”, as introduced by Mumford, using only the tools of algebraic topology.

Article information

Source
Geom. Topol., Volume 21, Number 4 (2017), 2015-2048.

Dates
Received: 30 October 2013
Revised: 25 May 2016
Accepted: 8 July 2016
First available in Project Euclid: 19 October 2017

Permanent link to this document
https://projecteuclid.org/euclid.gt/1508437636

Digital Object Identifier
doi:10.2140/gt.2017.21.2015

Mathematical Reviews number (MathSciNet)
MR3654103

Zentralblatt MATH identifier
1370.55005

Subjects
Primary: 55R40: Homology of classifying spaces, characteristic classes [See also 57Txx, 57R20] 55T10: Serre spectral sequences 57R22: Topology of vector bundles and fiber bundles [See also 55Rxx]

Keywords
manifold bundles characteristic classes tautological ring Miller–Morita–Mumford classes

Citation

Grigoriev, Ilya. Relations among characteristic classes of manifold bundles. Geom. Topol. 21 (2017), no. 4, 2015--2048. doi:10.2140/gt.2017.21.2015. https://projecteuclid.org/euclid.gt/1508437636


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References

  • J C Becker, D H Gottlieb, The transfer map and fiber bundles, Topology 14 (1975) 1–12
  • J M Boardman, Stable homotopy theory, V: Duality and Thom spectra, notes, University of Warwick (1966) Available at \setbox0\makeatletter\@url http://math.ucr.edu/~res/inprogress/Boardman-V.pdf {\unhbox0
  • S K Boldsen, Improved homological stability for the mapping class group with integral or twisted coefficients, Math. Z. 270 (2012) 297–329
  • A Borel, F Hirzebruch, Characteristic classes and homogeneous spaces, I, Amer. J. Math. 80 (1958) 458–538
  • C J Earle, J Eells, A fibre bundle description of Teichmüller theory, J. Differential Geometry 3 (1969) 19–43
  • J Ebert, O Randal-Williams, Generalised Miller–Morita–Mumford classes for block bundles and topological bundles, Algebr. Geom. Topol. 14 (2014) 1181–1204
  • C Faber, A conjectural description of the tautological ring of the moduli space of curves, from “Moduli of curves and abelian varieties” (C Faber, E Looijenga, editors), Aspects Math. E33, Friedr. Vieweg, Braunschweig (1999) 109–129
  • B Farb, D Margalit, A primer on mapping class groups, Princeton Mathematical Series 49, Princeton University Press (2012)
  • S Galatius, O Randal-Williams, Homological stability for moduli spaces of high dimensional manifolds, I, preprint (2014)
  • S Galatius, O Randal-Williams, Stable moduli spaces of high-dimensional manifolds, Acta Math. 212 (2014) 257–377
  • J L Harer, Stability of the homology of the mapping class groups of orientable surfaces, Ann. of Math. 121 (1985) 215–249
  • A Hatcher, Algebraic topology, Cambridge University Press (2002)
  • E Looijenga, On the tautological ring of ${\mathcal{M}}_g$, Invent. Math. 121 (1995) 411–419
  • 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 McCleary, A user's guide to spectral sequences, 2nd edition, Cambridge Studies in Advanced Mathematics 58, Cambridge University Press (2001)
  • J W Milnor, J D Stasheff, Characteristic classes, Annals of Mathematics Studies 76, Princeton University Press (1974)
  • S Morita, Families of Jacobian manifolds and characteristic classes of surface bundles, I, Ann. Inst. Fourier $($Grenoble$)$ 39 (1989) 777–810
  • S Morita, Generators for the tautological algebra of the moduli space of curves, Topology 42 (2003) 787–819
  • D Mumford, Towards an enumerative geometry of the moduli space of curves, from “Arithmetic and geometry, II” (M Artin, J Tate, editors), Progr. Math. 36, Birkhäuser, Boston (1983) 271–328
  • R Pandharipande, A Pixton, Relations in the tautological ring of the moduli space of curves, preprint (2013)
  • O Randal-Williams, Relations among tautological classes revisited, Adv. Math. 231 (2012) 1773–1785
  • N E Steenrod, Homology with local coefficients, Ann. of Math. 44 (1943) 610–627
  • R Thom, Quelques propriétés globales des variétés différentiables, Comment. Math. Helv. 28 (1954) 17–86