# What is a finite Haken cover of the Seifert–Weber space?

It's known that the Seifert–Weber space (obtained from a dodecahedron by gluing opposite faces with a 3/10 turn) is an example of a non-Haken 3-manifold. Since every closed 3-manifold is virtually Haken, I was wondering: is there was a known finite cover of the Seifert-Weber space that is Haken?

• It's not clear to me immediately if there are any smaller covers than Hempel's 5-sheeted one. But in principle you could turn this into a fairly simple algorithmic problem by iterating over the covering spaces and applying Reidemeister-Schrier to the corresponding subgroups. – Ryan Budney Jan 18 '17 at 23:14
• There are no covers of index $<5$, since the homology is $Z/5^3$ (hence any homomorphism to $S_4$ will be trivial). – Ian Agol Jan 18 '17 at 23:36

John Hempel, MR 664329 Orientation reversing involutions and the first Betti number for finite coverings of $3$-manifolds, Invent. Math. 67 (1982), no. 1, 133--142.
The Seifert-Weber space is a 5-fold cyclic branched cover over the Whitehead link complement. There are two such 5-fold covers (up to homeomorphism), which one may compute using SnapPea (perform $(5,0)$ surgery on each cusp of the Whitehead link, then compute all 5-fold cyclic covers of this orbifold, giving four manifold covers, with two isometry types). One may then compute the 5-fold cyclic covers of these two manifolds. One of them (not Seifert-Weber) has a 5-fold cyclic cover with positive betti number, whereas the 5-fold cyclic covers of the Seifert-Weber space have trivial betti number. However, one of them will be a 5-fold cover of its sibling, and hence will have a 5-fold cyclic cover which has positive betti number.
There are many other ways that we know the Seifert-Weber space to have a finite cover with positive first betti number (Hempel's paper pointed out by Igor shows that there is a 5-fold irregular cover), but it is an interesting question whether given a manifold $M$ with $b_1(M,\mathbb{F}_p)\geq 4$, is there a $p$-cover which has positive first betti number (I asked this as question 5 in a survey paper). Since all 5-fold covers of the Seifert-Weber space have $b_1(*; \mathbb{F}_5)\geq 4$, then this computation shows that a single 5-fold cyclic cover works in some cases.