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Thick isotopy property and the mapping class groups of Heegaard splittings
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Thick isotopy property and the mapping class groups of Heegaard splittings
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We give a necessary and sufficient condition for the fundamental group of the space of Heegaard splittings of an irreducible $3$-manifold to be finitely generated. The condition is exactly the conclusion of the thick isotopy lemma proved by Colding, Gabai and Ketover, which says that any isotopy of a Heegaard surface is achieved by a $1$-parameter family of surfaces with area bounded above by a universal constant and with some ``thickness property''. We also prove that a Heegaard splitting of a hyperbolic or spherical $3$-manifold satisfies the condition if it is topologically minimal (in the sense of Bachman) and its disk complex has finitely generated homotopy group. In conclusion, such a Heegaard splitting has finitely generated mapping class group.
Forward citations
Cited by 2 Pith papers
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A proof of Powell's conjecture on the Goeritz group of $S^3$
Proves Powell's conjecture that the Goeritz group for genus g≥3 Heegaard splittings of S³ is generated by four elements, using the topological minimality of the Heegaard surface.
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A proof of Powell's conjecture on the Goeritz group of $S^3$
Proves that the Goeritz group of genus g≥3 Heegaard splittings of S^3 is generated by four elements, using topological minimality.
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