Mapping class group and the Casson invariant

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 1107-1138
  • ISSN: 0373-0956

Abstract

top
Using a new definition of the second and third Johsnon homomorphisms, we simplify and extend the work of Morita on the Casson invariant of homology-spheres defined by Heegard splittings. In particular, we calculate the Casson invariant of the homology-sphere obtained by gluing two handlebodies along a homeomorphism of the boundary belonging to the Torelli subgroup.

How to cite

top

Perron, Bernard. "Mapping class group and the Casson invariant." Annales de l’institut Fourier 54.4 (2004): 1107-1138. <http://eudml.org/doc/116132>.

@article{Perron2004,
abstract = {Using a new definition of the second and third Johsnon homomorphisms, we simplify and extend the work of Morita on the Casson invariant of homology-spheres defined by Heegard splittings. In particular, we calculate the Casson invariant of the homology-sphere obtained by gluing two handlebodies along a homeomorphism of the boundary belonging to the Torelli subgroup.},
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; Johnson-Morita homomorphisms; homology spheres; Casson invariant},
language = {eng},
number = {4},
pages = {1107-1138},
publisher = {Association des Annales de l'Institut Fourier},
title = {Mapping class group and the Casson invariant},
url = {http://eudml.org/doc/116132},
volume = {54},
year = {2004},
}

TY - JOUR
AU - Perron, Bernard
TI - Mapping class group and the Casson invariant
JO - Annales de l’institut Fourier
PY - 2004
PB - Association des Annales de l'Institut Fourier
VL - 54
IS - 4
SP - 1107
EP - 1138
AB - Using a new definition of the second and third Johsnon homomorphisms, we simplify and extend the work of Morita on the Casson invariant of homology-spheres defined by Heegard splittings. In particular, we calculate the Casson invariant of the homology-sphere obtained by gluing two handlebodies along a homeomorphism of the boundary belonging to the Torelli subgroup.
LA - eng
KW - mapping class group; Johnson-Morita homomorphisms; homology spheres; Casson invariant
UR - http://eudml.org/doc/116132
ER -

References

top
  1. J. Birman, Braids, links and mapping class groups, vol 82 (1974), Princeton Univ. Press, Princeton Zbl0305.57013MR375281
  2. A. Casson, Lectures at MSRI (1985) 
  3. J.-M. Gambaudo, E. Ghys, Braids and Signatures Zbl1103.57001
  4. C.M.cA. Gordon, R. Litherland, On the signature of a link, Invent. Math. 47 (1978), 53-69 Zbl0391.57004MR500905
  5. H.B. Griffiths, Automorphisms of a 3-dimensional handlebody, Abh. Math. Sem. Univ. Hamburg 26 (1964), 191-210 Zbl0229.57005MR159313
  6. L. Guillou, A. Marin, Notes sur l'invariant de Casson des sphères d'homologie de dimension 3, Enseign. math. 38 (1992), 233-290 Zbl0776.57008MR1189008
  7. D. Johnson, An abelian quotient of the mapping class group ℐ g , Math. Ann. 249 (1980), 225-242 Zbl0409.57009MR579103
  8. D. Johnson, The structure of the Torelli group I, Annals of Math. 118 (1983), 423-442 Zbl0549.57006MR727699
  9. D. Johnson, The structure of the Torelli group II, Topology 24 (1985), 113-126 Zbl0571.57009MR793178
  10. W. Meyer, Die signatur von lokalen koeffizientensystemen und Faserbündeln, Bonner Mathematische Schriften 53 (1972) Zbl0243.58004MR305402
  11. W. Meyer, Die signatur von Flächenbündeln, Math. Ann. 201 (1973), 239-264 Zbl0241.55019MR331382
  12. S. Morita, Casson's invariant for homology 3-spheres and characteristic classes of surface bundles I, Topology 28 (1989), 305-323 Zbl0684.57008MR1014464
  13. S. Morita, On the structure of the Torelli group and the Casson invariant, Topology 30 (1991), 603-621 Zbl0747.57010MR1133875
  14. S. Morita, The extension of Johnson's homomorphism from the Torelli group to the mapping class group, Invent. Math. 111 (1993), 197-224 Zbl0787.57008MR1193604
  15. 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
  16. S. Morita, Abelian quotients of subgroups of the mapping class group of surfaces, Duke Math. J. 70 (1993), 699-726 Zbl0801.57011MR1224104
  17. S. Morita, Casson invariant, signature defect of framed manifolds and second characteristic classes of surface bundles, J. Diff. Geom. 47 (1997), 560-599 Zbl0903.57008MR1617632
  18. D. Mullins, The generalized Casson invariant for 2-fold branched covers of § 3 and the Jones polynomial, Topology 32 (1993), 419-438 Zbl0784.57003MR1217078
  19. B. Perron, Homomorphic extensions of Johnson homomorphisms via Fox Calculus, Ann. Inst. Fourier 54 (2004), 1073-1106 Zbl1109.57013MR2111022
  20. W. Pitsch, Une construction intrinsèque du cÏur de l'invariant de Casson, Ann. Inst. Fourier 51 (2001), 1741-1761 Zbl0983.57010MR1871288
  21. B. Perron, J.-P. Vannier, Groupe de monodromie géométrique des singularités simples, Math. Ann. 306 (1996), 231-245 Zbl0863.32013MR1411346
  22. J. Powell, Two theorems on the mapping class group of surfaces, Proc. Amer. Math. Soc. 68 (1978), 347-350 Zbl0391.57009MR494115
  23. D. Rolfsen, Knots and links, (1976), Publish or Perish Inc., Berkeley CA Zbl0339.55004MR515288
  24. J. Singer, Three dimensional manifolds and their Heegard diagrams, Trans. Amer. Math. Soc. 35 (1933), 88-111 Zbl0006.18501MR1501673
  25. S. Suzuky, On homeomorphisms of a 3-dimensional handlebody, Can. J. Math. 29 (1977), 111-124 Zbl0339.57001MR433433

NotesEmbed ?

top

You must be logged in to post comments.