Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

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.

share|improve this question
add comment

2 Answers

up vote 14 down vote accepted

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.

share|improve this answer
    
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. –  Ian Agol Jan 27 '12 at 21:46
    
thanks for both of your answers –  yanqing Jan 28 '12 at 0:44
add comment

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.)

share|improve this answer
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.