×

zbMATH — the first resource for mathematics

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)
MSC:
57N10 Topology of general \(3\)-manifolds (MSC2010)
57M50 General geometric structures on low-dimensional manifolds
PDF BibTeX XML Cite
Full Text: DOI arXiv
References:
[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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.