# zbMATH — the first resource for mathematics

Locally unknotted spines of Heegaard splittings. (English) Zbl 1120.57007
The main result of this paper is the following: If $$(\Sigma, H_1, H_2)$$ is a strongly irreducible Heegaard splitting for a manifold $$M$$ then every spine of $$\Sigma$$ is locally unknotted. Thus the spines of the handlebodies of a strongly irreducible Heegaard splitting will intersect a closed ball in a graph which is isotopic into the boundary of the ball.

##### MSC:
 57M27 Invariants of knots and $$3$$-manifolds (MSC2010) 57Q10 Simple homotopy type, Whitehead torsion, Reidemeister-Franz torsion, etc.
##### Keywords:
spine; Heegard splitting; sweep-out
Full Text:
##### References:
 [1] A J Casson, C M Gordon, Reducing Heegaard splittings, Topology Appl. 27 (1987) 275 · Zbl 0632.57010 · doi:10.1016/0166-8641(87)90092-7 [2] J Cerf, Sur les difféomorphismes de la sphère de dimension trois $$(\Gamma_4=0)$$, Lecture Notes in Mathematics 53, Springer (1968) · Zbl 0164.24502 · doi:10.1007/BFb0060395 [3] H Rubinstein, M Scharlemann, Comparing Heegaard splittings of non-Haken 3-manifolds, Topology 35 (1996) 1005 · Zbl 0858.57020 · doi:10.1016/0040-9383(95)00055-0 [4] M Scharlemann, Local detection of strongly irreducible Heegaard splittings, Topology Appl. 90 (1998) 135 · Zbl 0926.57018 · doi:10.1016/S0166-8641(97)00184-3
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.