Homomorphic extensions of Johnson homomorphisms via Fox calculus

Bernard Perron[1]

  • [1] Université de Bourgogne, Institut de mathématiques de Bourgogne, UFR sciences et techniques, 9 avenue Alain Savary, BP 47870, 21078 Dijon cedex (France)

Annales de l’institut Fourier (2004)

  • Volume: 54, Issue: 4, page 1073-1106
  • ISSN: 0373-0956

Abstract

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Using Fox differential calculus, for any positive integer k , we construct a map on the mapping class group ℳ g , 1 of a surface of genus g with one boundary component, such that, when restricted to an appropriate subgroup, it coincides with the k + 1 t h Johnson-Morita homomorphism. This allows us to construct very easily a homomorphic extension to ℳ g , 1 of the second and third Johnson-Morita homomorphisms.

How to cite

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Perron, Bernard. "Homomorphic extensions of Johnson homomorphisms via Fox calculus." Annales de l’institut Fourier 54.4 (2004): 1073-1106. <http://eudml.org/doc/116131>.

@article{Perron2004,
abstract = {Using Fox differential calculus, for any positive integer $k$, we construct a map on the mapping class group $\{\mathcal \{M\}\}_\{g,1\}$ of a surface of genus $g$ with one boundary component, such that, when restricted to an appropriate subgroup, it coincides with the $ k+1th$ Johnson-Morita homomorphism. This allows us to construct very easily a homomorphic extension to $\{\mathcal \{M\}\}_\{g,1\}$ of the second and third Johnson-Morita homomorphisms.},
affiliation = {Université de Bourgogne, Institut de mathématiques de Bourgogne, UFR sciences et techniques, 9 avenue Alain Savary, BP 47870, 21078 Dijon cedex (France)},
author = {Perron, Bernard},
journal = {Annales de l’institut Fourier},
keywords = {mapping class group of a surface; Johnson-Morita homomorphisms; Fox differential calculus},
language = {eng},
number = {4},
pages = {1073-1106},
publisher = {Association des Annales de l'Institut Fourier},
title = {Homomorphic extensions of Johnson homomorphisms via Fox calculus},
url = {http://eudml.org/doc/116131},
volume = {54},
year = {2004},
}

TY - JOUR
AU - Perron, Bernard
TI - Homomorphic extensions of Johnson homomorphisms via Fox calculus
JO - Annales de l’institut Fourier
PY - 2004
PB - Association des Annales de l'Institut Fourier
VL - 54
IS - 4
SP - 1073
EP - 1106
AB - Using Fox differential calculus, for any positive integer $k$, we construct a map on the mapping class group ${\mathcal {M}}_{g,1}$ of a surface of genus $g$ with one boundary component, such that, when restricted to an appropriate subgroup, it coincides with the $ k+1th$ Johnson-Morita homomorphism. This allows us to construct very easily a homomorphic extension to ${\mathcal {M}}_{g,1}$ of the second and third Johnson-Morita homomorphisms.
LA - eng
KW - mapping class group of a surface; Johnson-Morita homomorphisms; Fox differential calculus
UR - http://eudml.org/doc/116131
ER -

References

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  13. S. Morita, The structure of the mapping class group and characteristic classes of surface bundles, Mapping class groups and Moduli spaces of Riemann surfaces 150 (1993), 303-315 Zbl0791.57018
  14. S. Morita, Abelian quotients of subgroups of the mapping class group of surfaces, Duke Math J. 70 (1993), 699-726 Zbl0801.57011MR1224104
  15. B. Perron, Mapping class group and the Casson invariant, Ann. Inst. Fourier 54 (2004), 1107-1138 Zbl1110.57011MR2111023
  16. B. Perron, J.-P. Vannier, Groupe de monodromie géométrique des singularités simples, Math. Ann 306 (1996), 231-245 Zbl0863.32013MR1411346
  17. J. Powell, Two theorems on the mapping class group of surfaces, Proc. AMS 68 (1978), 347-350 Zbl0391.57009MR494115
  18. N. Saveliev, Lectures on the topology of 3-manifolds. An introduction to the Casson Invariant, (1999), Berlin, New York Zbl0932.57001MR1712769
  19. S. Suzuky, On homeomorphisms of a 3-dimensional handlebody, Can. J. Math. 29 (1977), 111-124 Zbl0339.57001MR433433

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