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The L-space conjecture for closed irreducible 3-manifolds

Let MM be a closed, orientable, irreducible 33-manifold. An L-space is a rational homology 33-sphere whose Heegaard Floer homology has minimal possible rank.

L-space conjecture. The following statements are equivalent:

  1. MM is a non-L-space.
  2. π1(M)\pi_1(M) is left-orderable.
  3. MM admits a co-orientable taut foliation.

This conjecture relates Heegaard Floer homology, orderability of fundamental groups, and taut foliations. The paper verifies it for all surgeries on the (−2,3,2k+1)(-2,3,2k+1)-pretzel knot for k⩾3k\geqslant 3, while the equivalence remains open in general.

Equivalent formulations 1Other wordings

Other statements of this same problem, merged from separate entries. Each is equivalent to the statement above — proving any one settles them all.

  1. The L-space conjecture for closed irreducible 3-manifolds

    Let MM be a closed, orientable, and irreducible 33-manifold.

    L-space conjecture. The following statements are equivalent:

    1. MM is not a Heegaard Floer LL-space.
    2. MM admits a co-orientable taut foliation.
    3. π1(M)\pi_1(M) is left-orderable.

    This conjecture relates Heegaard Floer homology, taut foliations, and orderability of fundamental groups. The paper proves the equivalence for manifolds of Heegaard genus two; its status in the generality stated above is not established here.

    source: Tao Li, “Taut foliations of 3-manifolds with Heegaard genus two”, arXiv:2202.00737 (2023).

References

Primary source

Bojun Zhao, “Left-orderability in Dehn fillings of pseudo-Anosov mapping tori”, arXiv:2604.04629 (2026).

3-manifoldsheegaard-floer-homologyleft-orderabilitytaut-foliations
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