I am curious about how the Heegaard genus changes after a finite covering.

Is there anyone constructing an closed hyperbolic 3-manifold $N$ such that

the Heegaard genus of a finite covering of $N$ is less than the Heegaard genus of $N$?

Thank you!

Note: Heegaard genus of a 3-manifold means the minimal genus of all Heegaard splittings.


There are examples like this. Check out section 4.5 of Shalen's paper "Hyperbolic volume, Heegaard genus and ranks of groups." It's here: http://arxiv.org/abs/0904.0191

He gives a reference for a genus 3 example by Alan Reid and a sketch of a technique for producing examples by Hyam Rubinstein. Shalen also conjectures that the genus can drop by at most 1 in a finite cover of a closed hyperbolic 3-manifold.

  • $\begingroup$ I think there's a variation on Hyam's construction, where you can take a non-orientable manifold with a non-orientable Heegaard splitting, whose 2-fold orientation cover has an orientable splitting of smaller genus than a Heegaard splitting downstairs. $\endgroup$
    – Ian Agol
    Jan 27 '12 at 21:46

Hyam Rubinstein and me have results about the behavior of the Heegaard genus under double covers for non-Haken manifolds, see http://arxiv.org/abs/math/0607145. Essentially, we show that the Heegaard genera of the two manifolds bound each other linearly. (The statement is a little more complicated for branched covers.)


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