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Thin position of Heegaard splittings. (English) Zbl 0930.57016
Since the concept of thin position for knots in the 3-sphere was developed by D. Gabai [J. Differ. Geom. 26, 479-536 (1987; Zbl 0639.57008)], many of its applications dealing with some problems related to incompressible surfaces and Heegaard splittings have been successfully made [see, for example, A. Thompson, Math. Res. Lett. 1, No. 5, 613-630 (1994; Zbl 0849.57009), Topology 36, No. 2, 505-507 (1997; Zbl 0867.57009); M. Scharlemann and A. Thompson, Math. Ann. 295, No. 3, 549-564 (1993; Zbl 0814.57010), Contemp. Math. 164, 231-238 (1994; Zbl 0818.57013), J. Differ. Geom. 39, No. 2, 343-357 (1994; Zbl 0820.57005)]. In this paper the authors define an analogous concept of thin position for Heegaard splittings, and discuss its relation to reducibilities of Heegaard splittings.
##### MSC:
 57N10 Topology of general $$3$$-manifolds (MSC2010)
reducibility