Algebraic & Geometric Topology

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

Matthew B Day

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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 k≥2, 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
    Mathematical Reviews (MathSciNet): MR0188307
    Zentralblatt MATH: 0135.04502
    Digital Object Identifier: doi:10.1112/plms/s3-15.1.239
  • K S Brown, Cohomology of groups, Graduate Texts in Mathematics 87, Springer, New York (1982)
    Mathematical Reviews (MathSciNet): MR672956
    Zentralblatt MATH: 0584.20036
  • R M Hain, Higher Albanese manifolds, from: “Hodge theory (Sant Cugat, 1985)”, Lecture Notes in Math. 1246, Springer, Berlin (1987) 84–91
    Mathematical Reviews (MathSciNet): MR894044
    Zentralblatt MATH: 0646.14017
  • 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
    Mathematical Reviews (MathSciNet): MR1234261
  • 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
    Mathematical Reviews (MathSciNet): MR1397061
    Zentralblatt MATH: 0868.14006
  • R M Hain, Infinitesimal presentations of the Torelli groups, J. Amer. Math. Soc. 10 (1997) 597–651
    Mathematical Reviews (MathSciNet): MR1431828
    Zentralblatt MATH: 0915.57001
    Digital Object Identifier: doi:10.1090/S0894-0347-97-00235-X
  • R M Hain, S Zucker, Unipotent variations of mixed Hodge structure, Invent. Math. 88 (1987) 83–124
    Mathematical Reviews (MathSciNet): MR877008
    Zentralblatt MATH: 0622.14007
    Digital Object Identifier: doi:10.1007/BF01405093
  • A Heap, Bordism invariants of the mapping class group, Topology 45 (2006) 851–886
    Mathematical Reviews (MathSciNet): MR2248515
    Zentralblatt MATH: 1156.57017
    Digital Object Identifier: doi:10.1016/j.top.2006.06.001
  • M W Hirsch, Differential topology, Graduate Texts in Mathematics 33, Springer, New York (1994)
    Mathematical Reviews (MathSciNet): MR1336822
  • K Igusa, K E Orr, Links, pictures and the homology of nilpotent groups, Topology 40 (2001) 1125–1166
    Mathematical Reviews (MathSciNet): MR1867241
    Zentralblatt MATH: 1002.57012
    Digital Object Identifier: doi:10.1016/S0040-9383(00)00002-1
  • D Johnson, An abelian quotient of the mapping class group $\mathcal{I}_g$, Math. Ann. 249 (1980) 225–242
    Mathematical Reviews (MathSciNet): MR579103
    Digital Object Identifier: doi:10.1007/BF01363897
  • 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
    Mathematical Reviews (MathSciNet): MR718141
    Zentralblatt MATH: 0553.57002
  • N Kawazumi, Cohomological aspects of Magnus expansions
    arXiv: math.GT/0505497
  • N Kawazumi, Harmonic Magnus expansion on the universal family of Riemann surfaces
    arXiv: math.GT/0603158
  • J-L Koszul, Homologie et cohomologie des algèbres de Lie, Bull. Soc. Math. France 78 (1950) 65–127
    Mathematical Reviews (MathSciNet): MR0036511
    Digital Object Identifier: doi:10.24033/bsmf.1410
  • S Morita, Abelian quotients of subgroups of the mapping class group of surfaces, Duke Math. J. 70 (1993) 699–726
    Mathematical Reviews (MathSciNet): MR1224104
    Zentralblatt MATH: 0801.57011
    Digital Object Identifier: doi:10.1215/S0012-7094-93-07017-2
    Project Euclid: euclid.dmj/1077290894
  • S Morita, The extension of Johnson's homomorphism from the Torelli group to the mapping class group, Invent. Math. 111 (1993) 197–224
    Mathematical Reviews (MathSciNet): MR1193604
    Digital Object Identifier: doi:10.1007/BF01231286
  • 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
    Mathematical Reviews (MathSciNet): MR1659679
    Zentralblatt MATH: 0939.32011
  • S Morita, R C Penner, Torelli groups, extended Johnson homomorphisms and new cycles on the moduli space of curves
    arXiv: math/0602461
    Zentralblatt MATH: 1157.32015
    Digital Object Identifier: doi:10.1017/S0305004107000990
  • K Nomizu, On the cohomology of compact homogeneous spaces of nilpotent Lie groups, Ann. of Math. $(2)$ 59 (1954) 531–538
    Mathematical Reviews (MathSciNet): MR0064057
    Digital Object Identifier: doi:10.2307/1969716
  • B Perron, Homomorphic extensions of Johnson homomorphisms via Fox calculus, Ann. Inst. Fourier $($Grenoble$)$ 54 (2004) 1073–1106
    Mathematical Reviews (MathSciNet): MR2111022
    Digital Object Identifier: doi:10.5802/aif.2044
  • M S Raghunathan, Discrete subgroups of Lie groups, Ergebnisse der Mathematik und ihrer Grenzgebiete 68, Springer, New York (1972)
    Mathematical Reviews (MathSciNet): MR0507234
    Zentralblatt MATH: 0254.22005
  • C A Weibel, An introduction to homological algebra, Cambridge Studies in Advanced Mathematics 38, Cambridge University Press, Cambridge (1994)
    Mathematical Reviews (MathSciNet): MR1269324
    Zentralblatt MATH: 0797.18001

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