By a rational homology 3-sphere, I mean a compact oriented manifold three-manifold $Y$ with $H_1(Y)$ finite. My question is whether there exists a reasonable classification of such manifolds such that $\pi_1(Y)$ has a minimal presentation with two generators? How about three generators?

Here are the examples I know:

Example 1. Connected sum of lens spaces Example 2. Poincare sphere (Thanks to Qiaochu Yuan for pointing out in a previous version of this question that the Poincare sphere satisfies these criteria).

  • $\begingroup$ Well, there are other spherical 3-manifolds besides lens spaces and the Poincare dodecahedral sphere (en.wikipedia.org/wiki/Spherical_3-manifold), although I'm not sure which ones you get by taking connected sums of lens spaces. The prism manifolds admit a presentation with two generators. $\endgroup$ Jun 25, 2014 at 16:52
  • 1
    $\begingroup$ Two families: any nonzero rational surgery on a 2-bridge knot; any rational homology sphere with an open book with genus 1 and 1 boundary component (e.g. branched covers of the trefoil and the figure 8). These are special cases of rational homology spheres that admit a genus-2 Heegaard diagram. $\endgroup$ Jun 25, 2014 at 16:57
  • 2
    $\begingroup$ If I am not mistaken (!), this shows, together with the poincare conjecture, that $M$ is a connected sum of two lens spaces. So unless $M$ is a sum of lens spaces, it has $\pi_2 (M) =0$. $\endgroup$ Jun 25, 2014 at 17:37
  • 3
    $\begingroup$ A nice collection of examples comes from double branched covers $S^3$ branched along a 3-bridge knot. These all have Heegaard genus 2, and hence their fundamental group is generated by at most 2 elements. Moreover, the double branched cover of a knot is always a rational homology sphere. 3-manifolds with Heegaard genus 2 are always branched covers over the 3-sphere with branch set a knot or link. $\endgroup$ Jun 25, 2014 at 19:15
  • 1
    $\begingroup$ An interesting sidelight to this question is that a 3-manifold whose fundamental group is generated by 2 elements may have Heegaard genus greater than 2. Examples that are Seifert fibered spaces are due to Boileau-Zieschang (Invent. Math. 76 (1984), no. 3, 455–468), and more recently Tao Li (J. Amer. Math. Soc. 26 (2013), no. 3, 777–829) found hyperbolic examples. $\endgroup$ Jun 25, 2014 at 19:15

1 Answer 1


As indicated in the comments, the greatest class of rank 2 3-manifolds (including rational homology spheres) are the genus 2 manifolds, which as indicated by Ruberman are double branched covers over links (coming from the hyperelliptic involution of the genus 2 surface which extends over both handlebodies). This includes connect sums of lens spaces and all the spherical space forms, as you indicated. However, it was a celebrated result of Boileau and Zieschang that there are certain Seifert-fibered spaces with rank 2 fundamental group, but Heegaard genus $=3$. They completely classified the rank 2 Seifert fibered spaces. These examples were extended by Weidmann and Schultens-Weidmann to certain graph manifolds (with rank $2$ fundamental group and genus $>2$). I think the state of the art in the non-hyperbolic case is a partial classification theorem of Boileau-Weidmann.

It is still unknown whether 2-generated hyperbolic 3-manifolds have Heegaard genus 2. Thurston observed that such manifolds are also branched covers over a link in $S^3$. Also, certain finiteness results are known for hyperbolic manifolds with rank 2 fundamental group, genus $>2$, and injectivity radius bounded below which has been written by Souto.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

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