Steve Boyer, Cameron Gordon, and Liam Watson have an interesting new preprint out today on the arXiv. In it, they posit:
Conjecture. An irreducible rational homology 3-sphere is an L-space if and only if its fundamental group is not left-orderable.
The motivation here is as follows: An L-space is something whose Heegaard Floer homology is as simple as possible; such 3-manifolds have no taut foliations. A nice type of taut foliation are those that are R-covered, and in this case, the fundamental group of the 3-manifold inherits a left-order from the action of the leaf space. (I’m always assuming here that foliations are co-orientable.)
Of course, it’s not known whether every non-L-space has a taut foliation, and there are certainly non-R-covered foliations, so a reasonable initial reaction is that this conjecture isn’t very plausible. However, their paper outlines a surprising amount of evidence for it, and in this post I’ll give some more data in that direction.
For instance, Danny and I gave a list of 44 small hyperbolic 3-manifolds with non-left-orderable fundamental groups; it turns out all of these are indeed L-spaces.
More systematically, some years ago I ran some experiments on the 11,031 small hyperbolic 3-manifolds in the Hodgson-Weeks census and found:
- There are at least 205 that have left-orderable fundamental groups. These were certified by reps to PSL(2,R) whose euler class vanishes. Hence the rep lifts to an action on the line.
- At least 3038 (27%) of the census manifolds are L-spaces. These were certified using the exact triangle, starting with lens spaces and other elliptic manifolds as known examples of L-spaces.
Surprisingly, consistent with the conjecture, these two sets were disjoint, and the naive odds of this happening is 10-29!
Now another sort of order associated to every taut foliation is a circular order coming from its universal circle. Thus one could also ask:
Conjecture. An irreducible rational homology 3-sphere is an L-space if and only if its fundamental group is not circularly-orderable.
However, this conjecture is false; I found some 256 L-spaces in the above sample that had irreducible reps to PSL(2, R), and hence circular orders.