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Invariant Heegaard surfaces in manifolds with involutions and the Heegaard genus of double covers. (English) Zbl 1222.57014

Let \(M\) and \(N\) be closed orientable 3-manifolds, and \(\pi: M\rightarrow N\) a double cover, which may be branched. It is not difficult to lift a Heegaard surface of \(N\) to one of \(M\), and then to give an upper bound \(g(M)\leq 2g(N)+b-1\), where \(g(\cdot)\) denotes the Heegaard genus of a 3-manifold or the genus of a surface, and \(b\) is the bridge index of the branch set with respect to a minimal genus Heegaard surface for \(N\). In this case the branch set, if non-empty, is a link \(L\), and its bridge index is half the minimal number of arcs in which it can be cut off by the Heegaard surface such that in each complementary handlebody the sets of arcs are boundary parallel. The converse is more difficult and this is the problem treated in the paper under review. Note that any double cover is the quotient of an involution \(f:M\rightarrow M\).
In what follows let \(M\) denote an irreducible, orientable, atoroidal, non-Seifert fibered 3-manifold of Heegaard genus at least 2 admitting an orientation preserving involution \(f\) and a strongly irreducible Heegaard surface \(\Sigma\). Under these hypotheses it is shown that \(M\) has an invariant Heegaard surface of genus at most \(8g(\Sigma)-7\), and each complementary handlebody is invariant. This is used to show that if \(N\) is the quotient of the involution, i.e. there is a double cover \(\pi: M\rightarrow N\), then \(g(N)\leq 4g(\Sigma)-3\). It is also shown that \(b\leq 8g(\Sigma)-6\), where \(b\) is the bridge index of the branch set of the double cover with respect to the Heegaard surface constructed in \(N\). If \(M\) is a non-Haken manifold, then by the work of A. Casson and C. McA. Gordon [Topology Appl. 27, 275–283 (1987; Zbl 0632.57010)], any minimal Heegaard surface is in fact strongly irreducible. So for non-Haken manifolds the above inequalities can be stated in terms of \(g(M)\) instead of \(g(\Sigma)\). The authors use an invariant version of Cerf theory to prove that \(\Sigma\) and its image under \(f\), \(f(\Sigma)\), intersect nicely, and then use \(\Sigma\cup f(\Sigma)\) to construct a complex \(C\) with useful properties which is then used to construct an invariant Heegaard surface for \(M\). This technique is based on the work of H. Rubinstein and M. Scharlemann [Topology 35, No.4, 1005–1026 (1996; Zbl 0858.57020)].

MSC:

57M99 General low-dimensional topology
57N10 Topology of general \(3\)-manifolds (MSC2010)
57M27 Invariants of knots and \(3\)-manifolds (MSC2010)
57M10 Covering spaces and low-dimensional topology
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