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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

On the Jacobi group and the mapping class group of $S^3\times S^3$


Author: Nikolai A. Krylov
Journal: Trans. Amer. Math. Soc. 355 (2003), 99-117
MSC (2000): Primary 57R50, 57R52; Secondary 20J06
Published electronically: September 5, 2002
MathSciNet review: 1928079
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Abstract: The paper contains a proof that the mapping class group of the manifold $S^3\times S^3$ is isomorphic to a central extension of the (full) Jacobi group $\Gamma^J$by the group of 7-dimensional homotopy spheres. Using a presentation of the group $\Gamma^J$ and the $\mu$-invariant of the homotopy spheres, we give a presentation of this mapping class group with generators and defining relations. We also compute the cohomology of the group $\Gamma^J$ and determine 2-cocycles that correspond to the mapping class group of $S^3\times S^3$.


References [Enhancements On Off] (What's this?)

  • 1. R. Berndt and R. Schmidt: Elements of the Representation Theory of the Jacobi Group, Birkhäuser, Boston, 1998. MR 99i:11030
  • 2. J. Birman: On Siegel's modular group, Math. Ann. 191 (1971), 59-68. MR 43:6325
  • 3. W. Browder: Diffeomorphisms of 1-connected manifolds, Trans. Amer. Math. Soc. 128 (1967), 155-163. MR 35:3681
  • 4. W. Browder: Surgery on simply-connected manifolds, Springer-Verlag, New York, 1972. MR 50:11272
  • 5. K. Brown: Cohomology of Groups, Graduate Texts in Math., vol. 87, Springer-Verlag, New York, 1982. MR 83k:20002
  • 6. Y. Choie: A short note on the full Jacobi group, Proc. Amer. Math. Soc. 123 (1995), 2625-2627. MR 95k:11070
  • 7. J. Eells and N. Kuiper: An invariant for certain smooth manifolds, Annali di Math. 60 (1962), 93-110. MR 27:6280
  • 8. M. Eichler and D. Zagier: The Theory of Jacobi Forms, Birkhäuser, Boston, 1985. MR 86j:11043
  • 9. L. Evens: The Cohomology of Groups, Oxford Univ. Press, 1991. MR 93i:20059
  • 10. D. Fried: Word maps, isotopy and entropy, Trans. Amer. Math. Soc. 296 (1986), 851-859. MR 87k:58243
  • 11. G. Hochschild and J-P. Serre: Cohomology of group extensions, Trans. Amer. Math. Soc. 74 (1953), 110-134. MR 14:619b
  • 12. D. Johnson: A Survey of the Torelli Group, Contemporary Math. 20 (1983), 165-179. MR 85d:57009
  • 13. M. Kervaire and J. Milnor: Groups of homotopy spheres, Annals of Math. 77 (1963), 504-537. MR 26:5584
  • 14. M. Kreck: Isotopy classes of diffeomorphisms of $(k-1)-$connected almost parallelizable $2k-$manifolds, Algebraic Topology, Aarhus 1978, 643-663, Lecture Notes in Math., vol. 763, Springer, Berlin, 1979. MR 81i:57029
  • 15. B. Lawson and M. Michelson: Spin geometry, Princeton Math. Series, vol. 39, 1989. MR 91g:53001
  • 16. J. Milnor: Differentiable structures on spheres, American J. Math. 81 (1959), 962-972. MR 22:990
  • 17. J. Milnor: Lectures on the h-cobordism theorem, notes by L. Siebenmann and J. Sondow, Princeton University Press, 1965. MR 32:8352
  • 18. H. Sato: Diffeomorphism groups of $S^p\times S^q$and exotic spheres, Quart. J. Math. Oxford (ser. 2) 20 (1969), 255-276. MR 40:6584
  • 19. E. Turner: A survey of diffeomorphism groups. Algebraic and geometrical methods in topology. Lecture Notes in Math. 428, 200-219, Springer-Verlag, Berlin-Heidelberg-New York, 1974. MR 52:4321
  • 20. B. Wajnryb: Mapping class group of a surface is generated by two elements, Topology 35, No. 2 (1996), 377-383. MR 96m:57007

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Additional Information

Nikolai A. Krylov
Affiliation: Department of Mathematics, Statistics and Computer Science, University of Illinois at Chicago, 851 S. Morgan St., Chicago, Illinois 60607
Address at time of publication: School of Engineering and Science, International University Bremen, P. O. Box 750 561, 28725 Bremen, Germany
Email: krylov@math.uic.edu, n.krylov@iu-bremen.de

DOI: http://dx.doi.org/10.1090/S0002-9947-02-03051-9
PII: S 0002-9947(02)03051-9
Received by editor(s): July 18, 2001
Received by editor(s) in revised form: March 15, 2002
Published electronically: September 5, 2002
Article copyright: © Copyright 2002 American Mathematical Society