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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?

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    $\begingroup$ 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. $\endgroup$
    – Ryan Budney
    Commented Jan 18, 2017 at 23:14
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    $\begingroup$ 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
    Commented Jan 18, 2017 at 23:36

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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, 133--142.

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  • $\begingroup$ This is what I was looking for - thanks! $\endgroup$
    – user
    Commented Jan 18, 2017 at 21:33
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I just checked using SnapPea that there is a cover of the Seifert-Weber dodecahedral space of index 25 (a 5-fold cyclic cover of a 5-fold cyclic cover) which has positive first betti number hence is Haken.

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.

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