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 nonHaken 3manifold. Since every closed 3manifold is virtually Haken, I was wondering: is there was a known finite cover of the SeifertWeber space that is Haken?

1$\begingroup$ It's not clear to me immediately if there are any smaller covers than Hempel's 5sheeted one. But in principle you could turn this into a fairly simple algorithmic problem by iterating over the covering spaces and applying ReidemeisterSchrier to the corresponding subgroups. $\endgroup$ – Ryan Budney Jan 18 '17 at 23:14

2$\begingroup$ There are no covers of index $<5$, since the homology is $Z/5^3$ (hence any homomorphism to $S_4$ will be trivial). $\endgroup$ – Ian Agol Jan 18 '17 at 23:36
This is constructed (reasonably explicitly) in John Hempel's 1982 paper.
John Hempel, MR 664329 Orientation reversing involutions and the first Betti number for finite coverings of $3$manifolds, Invent. Math. 67 (1982), no. 1, 133142.
I just checked using SnapPea that there is a cover of the SeifertWeber dodecahedral space of index 25 (a 5fold cyclic cover of a 5fold cyclic cover) which has positive first betti number hence is Haken.
The SeifertWeber space is a 5fold cyclic branched cover over the Whitehead link complement. There are two such 5fold covers (up to homeomorphism), which one may compute using SnapPea (perform $(5,0)$ surgery on each cusp of the Whitehead link, then compute all 5fold cyclic covers of this orbifold, giving four manifold covers, with two isometry types). One may then compute the 5fold cyclic covers of these two manifolds. One of them (not SeifertWeber) has a 5fold cyclic cover with positive betti number, whereas the 5fold cyclic covers of the SeifertWeber space have trivial betti number. However, one of them will be a 5fold cover of its sibling, and hence will have a 5fold cyclic cover which has positive betti number.
There are many other ways that we know the SeifertWeber space to have a finite cover with positive first betti number (Hempel's paper pointed out by Igor shows that there is a 5fold 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 5fold covers of the SeifertWeber space have $b_1(*; \mathbb{F}_5)\geq 4$, then this computation shows that a single 5fold cyclic cover works in some cases.