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Surface complexes of Seifert fibered spaces. (English) Zbl 1469.57018

The curve complex of a surface has been extensively studied, it has relations with mapping class groups of surfaces and Heegaard splittings of 3-manifolds. In this paper this complex is generalized to the surface complex of a compact and orientable 3-manifold \(M\), denoted \(\mathcal{S}(M)\). This is defined inductively, via a sequence of complexes \(\mathcal{S}_0(M) \subset \mathcal{S}_1 (M)\subset \mathcal{S}_2 (M) \subset \cdots\) . Vertices of these complexes correspond to isotopy classes of compact connected orientable essential (incompressible, \(\partial\)-incompressible and non \(\partial\)-parallel) surfaces properly embedded in \(M\); edges in \(\mathcal{S}_0\) correspond to pairs of disjoint essential surfaces, and assuming that \(\mathcal{S}_d\) has been defined, additional edges in \(\mathcal{S}_{d+1}\) correspond to pairs of surfaces that lie in distinct components of \(\mathcal{S}_d\), and whose intersection is transverse and contain \({d+1}\) components. Then \(\mathcal{S}(M)\) is defined as \(\mathcal{S}(M)=\cup_i \mathcal{S}_i(M)\).
Let \(M\) be a Seifert fibered 3-manifold. It is well known that surfaces in \(M\) are either horizontal or vertical, that is, they are transverse to fibers of the Seifert fibration, or are made of fibers. By identifying each fiber of \(M\) to a point, an orbifold \(Q\), that is, a surface with cone points is obtained, where the cone points correspond to the exceptional fibers of \(M\). Let \(\hat Q\) be \(Q\) with an open regular neighborhood of the cone points removed. Vertical surfaces in \(M\) are mapped to essential curves in \(\hat Q\), so it is then natural to expect some relation between \(\mathcal{S}(M)\) and the curve complex of \(\hat Q\), \(\mathcal{C}(\hat Q)\). The main results of the paper are the following: Suppose that \(M\) is totally orientable (i.e., \(M\) and \(Q\) are orientable), (a) if the Euler number of \(M\) is not \(0\), then \(\mathcal{S}(M)\) is isomorphic to \(\mathcal{C}(\hat Q)\); (b) if \(M\) has Euler number \(0\) and the base orbifold has genus \(0\), then \(\mathcal{S}(M)\) has a subcomplex isomorphic to \(\mathcal{C}(\hat Q)\), and \(\mathcal{S}(M)\) is contained in the cone of this subcomplex; (c) if the Euler number of \(M\) is \(0\) and \(Q\) has positive genus, then \(\mathcal{S}(M)\) has a subcomplex isomorphic to \(\mathcal{C}(\hat Q)\), and \(\mathcal{S}_d(M)\) is connected for certain \(d\), and then \(\mathcal{S}_d(M)=\mathcal{S}(M)\).

MSC:

57K30 General topology of 3-manifolds
57K31 Invariants of 3-manifolds (including skein modules, character varieties)
57M50 General geometric structures on low-dimensional manifolds
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