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# Communications in Analysis and Geometry

## Volume 17 (2009)

### Number 5

### Invariant Heegaard surfaces in manifolds with involutions and the Heegaard genus of double covers

Pages: 851 – 901

DOI: http://dx.doi.org/10.4310/CAG.2009.v17.n5.a2

#### Authors

#### Abstract

Let $M$ be a 3-manifold admitting a strongly irreducibleHeegaard surface $\s$ and $f:M\to M$ an involution. Weconstruct an invariant Heegaard surface for $M$ of genus atmost $\bddup$. As a consequence, given a (possiblybranched) double cover $\pi:M \to N$ we obtain thefollowing bound on the Heegaard genus of $N$:\[g(N) \leq \bdddown.\]\noindent We also get a bound on the complexity of thebranch set in terms of $g(\s)$. If we assume that $M$ isnon-Haken, by Casson and Gordon \cite{casson-gordon} we mayreplace $g(\s)$ by $g(M)$ in all the statements above.