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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.

57N10 Topology of general \(3\)-manifolds (MSC2010)
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