Invariant Heegaard Surfaces in Manifolds with Involutions and the Heegaard Genus of Double Covers

Details Invariant Heegaard Surfaces in Manifolds with Involutions and the Heegaard Genus of Double Covers Invariant Heegaard Surfaces in Manifolds with Involutions and the Heegaard Genus of Double Covers

2006-07-06

arxiv.org/abs/math/0607145v2

Mathematics

Geometric Topology

Minor changes only. Final version, to appear in Communications in Analysis and Geometry

Scientific paper

Let M be a 3-manifold admitting a strongly irreducible Heegaard surface S and f:M \to M an involution. We construct an invariant Heegaard surface for M of genus at most 8 g(S) - 7. As a consequence, given a (possibly branched) double cover \pi:M \to N we obtain the following bound on the Heegaard genus of N: g(N) \leq 4g(S) - 3. We also get a bound on the complexity of the branch set in terms of g(S). If we assume that M is non-Haken, by Casson and Gordon we may replace g(S) by g(M) in all the statements above.

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