On the Jacobi group and the mapping class group of $S^3\times S^3$
HTML articles powered by AMS MathViewer
- by Nikolai A. Krylov
- Trans. Amer. Math. Soc. 355 (2003), 99-117
- DOI: https://doi.org/10.1090/S0002-9947-02-03051-9
- Published electronically: September 5, 2002
- PDF | Request permission
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
- Rolf Berndt and Ralf Schmidt, Elements of the representation theory of the Jacobi group, Progress in Mathematics, vol. 163, Birkhäuser Verlag, Basel, 1998. MR 1634977, DOI 10.1007/978-3-0348-0283-3
- Joan S. Birman, On Siegel’s modular group, Math. Ann. 191 (1971), 59–68. MR 280606, DOI 10.1007/BF01433472
- William Browder, Diffeomorphisms of $1$-connected manifolds, Trans. Amer. Math. Soc. 128 (1967), 155–163. MR 212816, DOI 10.1090/S0002-9947-1967-0212816-0
- William Browder, Surgery on simply-connected manifolds, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 65, Springer-Verlag, New York-Heidelberg, 1972. MR 0358813
- Kenneth S. Brown, Cohomology of groups, Graduate Texts in Mathematics, vol. 87, Springer-Verlag, New York-Berlin, 1982. MR 672956
- YoungJu Choie, A short note on the full Jacobi group, Proc. Amer. Math. Soc. 123 (1995), no. 9, 2625–2628. MR 1260164, DOI 10.1090/S0002-9939-1995-1260164-5
- James Eells Jr. and Nicolaas H. Kuiper, An invariant for certain smooth manifolds, Ann. Mat. Pura Appl. (4) 60 (1962), 93–110. MR 156356, DOI 10.1007/BF02412768
- Martin Eichler and Don Zagier, The theory of Jacobi forms, Progress in Mathematics, vol. 55, Birkhäuser Boston, Inc., Boston, MA, 1985. MR 781735, DOI 10.1007/978-1-4684-9162-3
- Leonard Evens, The cohomology of groups, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1991. Oxford Science Publications. MR 1144017
- David Fried, Word maps, isotopy and entropy, Trans. Amer. Math. Soc. 296 (1986), no. 2, 851–859. MR 846609, DOI 10.1090/S0002-9947-1986-0846609-X
- P. Hebroni, Sur les inverses des éléments dérivables dans un anneau abstrait, C. R. Acad. Sci. Paris 209 (1939), 285–287 (French). MR 14
- Dennis Johnson, A survey of the Torelli group, Low-dimensional topology (San Francisco, Calif., 1981) Contemp. Math., vol. 20, Amer. Math. Soc., Providence, RI, 1983, pp. 165–179. MR 718141, DOI 10.1090/conm/020/718141
- Michel A. Kervaire and John W. Milnor, Groups of homotopy spheres. I, Ann. of Math. (2) 77 (1963), 504–537. MR 148075, DOI 10.1090/S0273-0979-2015-01504-1
- M. Kreck, Isotopy classes of diffeomorphisms of $(k-1)$-connected almost-parallelizable $2k$-manifolds, Algebraic topology, Aarhus 1978 (Proc. Sympos., Univ. Aarhus, Aarhus, 1978), Lecture Notes in Math., vol. 763, Springer, Berlin, 1979, pp. 643–663. MR 561244
- H. Blaine Lawson Jr. and Marie-Louise Michelsohn, Spin geometry, Princeton Mathematical Series, vol. 38, Princeton University Press, Princeton, NJ, 1989. MR 1031992
- John Milnor, Differentiable structures on spheres, Amer. J. Math. 81 (1959), 962–972. MR 110107, DOI 10.2307/2372998
- John Milnor, Lectures on the $h$-cobordism theorem, Princeton University Press, Princeton, N.J., 1965. Notes by L. Siebenmann and J. Sondow. MR 0190942
- Hajime Sato, Diffeomorphism group of $S^{p}\times S^{q}$ and exotic spheres, Quart. J. Math. Oxford Ser. (2) 20 (1969), 255–276. MR 253369, DOI 10.1093/qmath/20.1.255
- Edward C. Turner, A survey of diffeomorphism groups, Algebraic and geometrical methods in topology (Conf. Topological Methods in Algebraic Topology, State Univ. New York, Binghamton, N.Y., 1973), Lecture Notes in Math., Vol. 428, Springer, Berlin, 1974, pp. 200–218. MR 0383440
- Bronislaw Wajnryb, Mapping class group of a surface is generated by two elements, Topology 35 (1996), no. 2, 377–383. MR 1380505, DOI 10.1016/0040-9383(95)00037-2
Bibliographic 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
- Received by editor(s): July 18, 2001
- Received by editor(s) in revised form: March 15, 2002
- Published electronically: September 5, 2002
- © Copyright 2002 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 355 (2003), 99-117
- MSC (2000): Primary 57R50, 57R52; Secondary 20J06
- DOI: https://doi.org/10.1090/S0002-9947-02-03051-9
- MathSciNet review: 1928079