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Transactions of the American Mathematical Society
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
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."


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DOI: http://dx.doi.org/10.1090/S0002-9947-1980-0549161-6
PII: S 0002-9947(1980)0549161-6
Keywords: Homeomorphism, deformation, isotopy, retract, handlebody, surface, graph, neighborhood, group, generator
Article copyright: © Copyright 1980 American Mathematical Society