Heegaard genus of the hyperbolic dodecahedral space (is it 3 or 4?) I have a question on the hyperbolic dodecahedral space, first described by C.Weber and H. Seifert in 1933 [Die beiden Dodekaederr\"aume, Math Z. 37 (1933), 237-253]. Is it known whether it admits a Heegaard splitting of genus 3? 
We can see (even with elementary combinatorial tools, like discrete Morse theory) that it does admit a Heegaard splitting of genus 4; but I was wondering if this is best possible. 
 A: The Whitehead link fibers with a twice-punctured genus 1 fiber.  Since the Seifert-Webber Space is a cyclic 5-fold covering of the Whitehead link, the image of Whitehead link in this cover should again be a fibered link with a twice-punctured genus 1 fiber. So this will give a genus 3 Heegaard splitting.  
I reckon this argument should work to show each of the two isometry classes of 5-fold cyclic covers (as mentioned in the comments to this question) have genus 3 Heegaard splittings.
Edit:  Of course Agol's comment is right.  I wasn't considering the other possible 5-fold cyclic branched covers.  
So if I've got my head on straight about it, these other covers should be dual to a thrice-punctured genus 1 fiber of the link exterior.  (Such a fiber is obtained from adding the twice-punctured torus Seifert surface with a suitably oriented once punctured torus bounded by one component and disjoint from the other.)  Then in the cover, as well as in S^3, the two boundary components of the fiber on the same link component join together to make a non-orientable surface with $\chi = -3$ and one boundary component.  The boundary of a neighborhood of this surface then gives a genus 4 Heegaard splitting.  Presumably, this splitting is equivalent to the ones Bruno obtained.
