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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Homeomorphisms of $ S\sp{3}$ leaving a Heegaard surface invariant


Author: Jerome Powell
Journal: Trans. Amer. Math. Soc. 257 (1980), 193-216
MSC: Primary 57N10
DOI: https://doi.org/10.1090/S0002-9947-1980-0549161-6
MathSciNet review: 549161
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Abstract: We find a finite set of generators for the group $ {\mathcal{H} _g}$ of isotopy classes of orientation-preserving homeomorphisms of the 3-sphere $ {S^3}$ which leave a Heegaard surface T of genus g in $ {S^3}$ invariant. We also show that every element of the group $ {\mathcal{H} _g}$ can be represented by a deformation of the surface T in $ {S^3}$ of a very special type: during the deformation the surface T is the boundary of the regular neighborhood of a graph embedded in a fixed 2-sphere. The only exception occurs when a subset of the graph contained in a disc on the 2-sphere is ``flipped over."


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

  • [B-1] J. Birman, Braids, links and mapping class groups, Ann. of Math Studies, no. 82, Princeton Univ. Press, Princeton, N. J. 1974. MR 0375281 (51:11477)
  • [B-P] J. Birman and J. Powell, Special representations of 3-manifolds (to appear). MR 537723 (80g:57010)
  • [C] R. Craggs, Stable representations for 3-and 4-manifolds (preprint).
  • [G] L. Goeritz, Die Abbildungen der Brezelflache und der Volbrezel vom Geschlect 2, Abh. Math. Sem. Univ. Hamburg 9 (1933), 244-259.
  • [Ha] M.-E. Hamstrom, The space of homeomorphisms on a torus, Illinois J. Math. 9 (1965), 59-65. MR 0170334 (30:572)
  • [Hi] H. Hilden, Generators for two groups related to the braid group, Pacific J. Math. 59 (1975), 475-486. MR 0388372 (52:9209)
  • [M] D. R. McMillan, Jr., Homeomorphisms on a solid torus, Proc. Amer. Math. Soc. 14 (1963), 386-390. MR 0150783 (27:770)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1980-0549161-6
Keywords: Homeomorphism, deformation, isotopy, retract, handlebody, surface, graph, neighborhood, group, generator
Article copyright: © Copyright 1980 American Mathematical Society

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