Algebraic & Geometric Topology

Extending Johnson's and Morita's homomorphisms to the mapping class group

Matthew B Day

Full-text: Open access

Abstract

We extend certain homomorphisms defined on the higher Torelli subgroups of the mapping class group to crossed homomorphisms defined on the entire mapping class group. In particular, for every k2, we construct a crossed homomorphism ϵk which extends Morita’s homomorphism τ̃k to the entire mapping class group. From this crossed homomorphism we also obtain a crossed homomorphism extending the kth Johnson homomorphism τk to the mapping class group.

D Johnson and S Morita obtained their respective homomorphisms by considering the action of the mapping class group on the nilpotent truncations of the surface group; our approach is to mimic Morita’s construction topologically by using nilmanifolds associated to these truncations. This allows us to take the ranges of these crossed homomorphisms to be certain finite-dimensional real vector spaces associated to these nilmanifolds.

Article information

Source
Algebr. Geom. Topol., Volume 7, Number 3 (2007), 1297-1326.

Dates
Received: 26 February 2007
Revised: 3 August 2007
Accepted: 15 August 2007
First available in Project Euclid: 20 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.agt/1513796742

Digital Object Identifier
doi:10.2140/agt.2007.7.1297

Mathematical Reviews number (MathSciNet)
MR2350283

Zentralblatt MATH identifier
1181.57025

Subjects
Primary: 57N05: Topology of $E^2$ , 2-manifolds
Secondary: 57T15: Homology and cohomology of homogeneous spaces of Lie groups

Keywords
mapping class group Johnson homomorphism Torelli group

Citation

Day, Matthew B. Extending Johnson's and Morita's homomorphisms to the mapping class group. Algebr. Geom. Topol. 7 (2007), no. 3, 1297--1326. doi:10.2140/agt.2007.7.1297. https://projecteuclid.org/euclid.agt/1513796742


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References

  • S Andreadakis, On the automorphisms of free groups and free nilpotent groups, Proc. London Math. Soc. $(3)$ 15 (1965) 239–268
  • K S Brown, Cohomology of groups, Graduate Texts in Mathematics 87, Springer, New York (1982)
  • R M Hain, Higher Albanese manifolds, from: “Hodge theory (Sant Cugat, 1985)”, Lecture Notes in Math. 1246, Springer, Berlin (1987) 84–91
  • R M Hain, Completions of mapping class groups and the cycle $C{-}C^-$, from: “Mapping class groups and moduli spaces of Riemann surfaces (Göttingen, 1991/Seattle, WA, 1991)”, Contemp. Math. 150, Amer. Math. Soc., Providence, RI (1993) 75–105
  • R M Hain, Torelli groups and geometry of moduli spaces of curves, from: “Current topics in complex algebraic geometry (Berkeley, CA, 1992/93)”, Math. Sci. Res. Inst. Publ. 28, Cambridge Univ. Press, Cambridge (1995) 97–143
  • R M Hain, Infinitesimal presentations of the Torelli groups, J. Amer. Math. Soc. 10 (1997) 597–651
  • R M Hain, S Zucker, Unipotent variations of mixed Hodge structure, Invent. Math. 88 (1987) 83–124
  • A Heap, Bordism invariants of the mapping class group, Topology 45 (2006) 851–886
  • M W Hirsch, Differential topology, Graduate Texts in Mathematics 33, Springer, New York (1994)
  • K Igusa, K E Orr, Links, pictures and the homology of nilpotent groups, Topology 40 (2001) 1125–1166
  • D Johnson, An abelian quotient of the mapping class group $\mathcal{I}_g$, Math. Ann. 249 (1980) 225–242
  • D Johnson, A survey of the Torelli group, from: “Low-dimensional topology (San Francisco, CA, 1981)”, Contemp. Math. 20, Amer. Math. Soc., Providence, RI (1983) 165–179
  • N Kawazumi, Cohomological aspects of Magnus expansions
  • N Kawazumi, Harmonic Magnus expansion on the universal family of Riemann surfaces
  • J-L Koszul, Homologie et cohomologie des algèbres de Lie, Bull. Soc. Math. France 78 (1950) 65–127
  • S Morita, Abelian quotients of subgroups of the mapping class group of surfaces, Duke Math. J. 70 (1993) 699–726
  • S Morita, The extension of Johnson's homomorphism from the Torelli group to the mapping class group, Invent. Math. 111 (1993) 197–224
  • S Morita, A linear representation of the mapping class group of orientable surfaces and characteristic classes of surface bundles, from: “Topology and Teichmüller spaces (Katinkulta, 1995)”, World Sci. Publ., River Edge, NJ (1996) 159–186
  • S Morita, R C Penner, Torelli groups, extended Johnson homomorphisms and new cycles on the moduli space of curves
  • K Nomizu, On the cohomology of compact homogeneous spaces of nilpotent Lie groups, Ann. of Math. $(2)$ 59 (1954) 531–538
  • B Perron, Homomorphic extensions of Johnson homomorphisms via Fox calculus, Ann. Inst. Fourier $($Grenoble$)$ 54 (2004) 1073–1106
  • M S Raghunathan, Discrete subgroups of Lie groups, Ergebnisse der Mathematik und ihrer Grenzgebiete 68, Springer, New York (1972)
  • C A Weibel, An introduction to homological algebra, Cambridge Studies in Advanced Mathematics 38, Cambridge University Press, Cambridge (1994)