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



Yo’av Rieck (Department of Mathematical Sciences, University of Arkansas)

J. Hyam Rubinstein (Department of Mathematics and Statistics, University of Melbourne, Australia)


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.

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