Heegaard splittings of \((\text{surface})\times I\) are standard. (English) Zbl 0814.57010

C. Frohman and J. Hass have shown [Invent. Math. 95, No. 3, 529-540 (1989; Zbl 0678.57009)] that genus three Heegaard splittings of the 3-torus are standard. M. Boileau and J.-P. Otal [J. Differ. Geom. 32, No. 1, 209-233 (1990; Zbl 0754.53012)] generalize this result to show that all Heegaard splittings of the 3-torus are standard. A crucial part of Boileau-Otal’s argument is to show that all Heegaard splittings of a torus crossed with an interval are standard. We generalize this part of their paper to prove that all Heegaard splittings of a closed orientable genus \(g\) surface crossed with an interval are standard. Many of our arguments are based on theirs; we differ substantially in Sect. 4, which allows us to obtain the more general result.
The paper is organized as follows: Section 1 begins with a discussion of compression bodies and their spines. In Sect. 2 we discuss Heegaard splittings and state the main theorem (2.11). The proof of the main theorem begins in Sect. 3, where we prove a lemma which splits the remainder of the proof into two cases. These cases are considered in Sect. 4 and Sect. 5. In Sect. 6 we exploit the main theorem to generalize a theorem of Frohman.
Pitts and Rubinstein have announced related results, using different methods.


57N10 Topology of general \(3\)-manifolds (MSC2010)
57Q37 Isotopy in PL-topology
57Q35 Embeddings and immersions in PL-topology
57M15 Relations of low-dimensional topology with graph theory
Full Text: DOI EuDML


[1] [BO] Boileau, M., Otal, J.-P.: Sur les scindements de Heegaard du toreT 3. J. Diff. Geom.32, 209-233 (1990) · Zbl 0754.53012
[2] [CG] Casson, A., Gordon, C. McA.: Reducing Heegaard splittings. Topology Appl.27, 275-283 (1990) · Zbl 0632.57010
[3] [F] Frohman, C.: The topological uniqueness of triply periodic minimal surfaces inR 3. J. Differ. Geom.31, 277-283 (1990) · Zbl 0689.53002
[4] [F2] Frohman, C.: An unknotting lemma for systems of arcs inFxI. Pac. J. Math.139, 59-66 (1989) · Zbl 0693.57010
[5] [FH] Frohman, C., Hass, J.: Unstable minimal surfaces and Heegaard splittings. Invent. Math.95, 529-540 (1989) · Zbl 0678.57009
[6] [G] Gabai, D.: Foliations and the topology of 3-manifolds III. J. Differ. Geom.26, 479-536 (1987) · Zbl 0639.57008
[7] [Go] Gordon, C.: On primitive sets of loops in the boundary of a handlebody. Top. Appl. Phys.27, 285-299 (1987) · Zbl 0634.57007
[8] [H] Haken, W.: Some results on surfaces in 3-manifolds (Studies in Modern Topology, M. A. A., pp. 34-98) Englewood Cliffs: Prentice Hall 1968
[9] [J] Jaco, W.: Adding a 2-handle to a 3-manifold: an application to propertyR. Proc. Am. Math. Soc.92, 288-292 (1984) · Zbl 0564.57009
[10] [ST] Scharlemann, M., Thompson, A.: Detecting unknotted graphs in 3-space. J. Differ. Geom.34, 539-560 (1991) · Zbl 0751.05033
[11] [W] Waldhausen, F.: Heegaard-Zerlegungen der 3-Sph?re. Topology7, 195-203 (1968) · Zbl 0157.54501
[12] [Wu] Wu, Y.-Q.: A generalization of the handle addition theorem. Proc. Amer. Math. Soc.114, 237-242 (1992) · Zbl 0737.57007
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.