# Question about the fundamental group of rational homology 3-spheres

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).

• 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. Jun 25, 2014 at 16:52
• 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. Jun 25, 2014 at 16:57
• 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 $\pi_2 (M) =0$. Jun 25, 2014 at 17:37
• 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. Jun 25, 2014 at 19:15
• 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. Jun 25, 2014 at 19:15

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.