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3-manifolds as viewed from the curve complex. (English) Zbl 0985.57014
The author studies topological properties of 3-manifolds and their splittings in terms of the geometry and combinatorics of the associated curve complexes. A curve complex $$C(S)$$ of a closed, connected, oriented surface $$S$$ of genus $$g\geq 2$$ is the complex whose vertices are the isotopy classes of essential simple closed curves of $$S$$, and where $$k+1$$ distinct vertices determine a $$k$$-simplex if they are represented by pairwise disjoint simple closed curves. If $$S$$ is endowed with an hyperbolic metric, each isotopy class contains a unique geodesics; so the vertices of $$C(S)$$ can be considered as geodesics. The geodesic distance function $$d$$ is defined on the 0-skeleton of $$C(S)$$ by $$d(x,y)=$$ the minimal number of 1-simplexes in a simplicial path joining $$x$$ to $$y$$. A pair $$X,Y$$ of simplexes of $$C(S)$$ determines a Heegaard splitting of a 3-manifold for which $$(S;X,Y)$$ is a Heegaard diagram. A certain pair of subcomplexs $$K_X,K_Y\subset C(S)$$, associated to $$X$$ and $$Y$$, describes the different diagrams for a fixed splitting. The distance of the splitting $$d(K_X,K_Y)$$ is the minimal distance between their respective vertices.
The paper shows that there are distance $$n$$ splittings of closed, oriented 3-manifolds for arbitrary large $$n$$. But any splitting of a 3-manifold which is Seifert fibered or which contains an essential torus has distance $$\leq 2$$. The author also gives a criterion for recognizing distance three splittings and describes all 3-manifolds which admit distance two, genus two splittings.

MSC:
 57N10 Topology of general $$3$$-manifolds (MSC2010)
Keywords:
Heegaard splitting
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