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How a strongly irreducible Heegaard splitting intersects a handlebody. (English) Zbl 0974.57011
The notion of a strongly irreducible Heegaard splitting was introduced by A. J. Casson and C. McA. Gordon [ibid. 27, 275-283 (1987; Zbl 0632.57010)]. In the paper under review, the authors show how a strongly irreducible Heegaard splitting surface \(Q\) of a 3-manifold \(M\) with extra side conditions intersects an arbitrary genus handlebody \(H\) in \(M\). In a previous paper [ibid. 90, No. 1-3, 135-147 (1998; Zbl 0926.57018)], the second author investigated the case when \(H\) is either a ball or solid torus. The side conditions imply that the surface is weakly incompressible, so that the problem becomes a problem in characterizing weakly incompressible surfaces embedded in a handlebody.
Reviewer: J.Hebda (St.Louis)
57N10 Topology of general \(3\)-manifolds (MSC2010)
57M50 General geometric structures on low-dimensional manifolds
Full Text: DOI arXiv
[1] Casson, A.; Gordon, C.McA., Reducing Heegaard splittings, Topology appl., 27, 275-283, (1987) · Zbl 0632.57010
[2] Frohman, C., Minimal surfaces and Heegaard splittings of the three-torus, Pacific J. math., 124, 119-130, (1986) · Zbl 0604.57006
[3] Jaco, W., Lectures on three-manifold topology, (1980), Amer. Math. Soc. Providence, RI · Zbl 0433.57001
[4] Rubinstein, H.; Scharlemann, M., Comparing Heegaard splittings of non-haken 3-manifolds, Topology, 35, 1005-1026, (1996) · Zbl 0858.57020
[5] Rubinstein, H.; Scharlemann, M., Comparing Heegaard splittings—the bounded case, Trans. amer. math. soc., 350, 689-715, (1998) · Zbl 0892.57009
[6] Rubinstein, H.; Scharlemann, M., Genus two Heegaard splittings of orientable \(3\)-manifolds, () · Zbl 0962.57013
[7] Scharlemann, M., Local detection of strongly irreducible Heegaard splittings, Topology appl., 90, 135-147, (1998) · Zbl 0926.57018
[8] M. Scharlemann, Heegaard splittings of compact \(3\)-manifolds, in: R. Daverman, R. Sher (Eds.), Handbook of Geometric Topology, Elsevier, Amsterdam, to appear · Zbl 0985.57005
[9] Scharlemann, M.; Thompson, A., Thin position and Heegaard splittings of the 3-sphere, J. differential geom., 39, 343-357, (1994) · Zbl 0820.57005
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