On 3-manifolds admitting co-orientable taut foliations, but none with vanishing Euler class

In this article, we construct infinitely many (small Seifert fibred, hyperbolic and toroidal) rational homology $3$-spheres that admit co-orientable taut foliations, but none with vanishing Euler class. In the context of the $L$-space conjecture, these examples provide rational homology $3$-spheres that admit co-orientable taut foliations (and hence are not $L$-spaces) and have left-orderable fundamental groups, yet none of the left orders arise directly from the universal circle actions associated to co-orientable taut foliations. The hyperbolic and non-Seifert toroidal examples are obtained from Dehn surgeries on knots in the $3$-sphere and use Heegaard Floer homology to obstruct the existence of a co-orientable foliation with vanishing Euler class. For the Seifert fibred case, we establish necessary and sufficient conditions for the Euler class of the normal bundle of the Seifert fibration to vanish. Moreover, when the base orbifold is hyperbolic, we also provide a second proof of this condition from the viewpoint of discrete faithful representations of Fuchsian groups.

💡 Research Summary

The paper addresses the long‑standing question of whether every rational homology 3‑sphere that admits a co‑oriented taut foliation also admits one whose Euler class vanishes. While the answer is trivially “yes” when the second homology consists only of 2‑torsion, no counter‑examples were known in the general case. The authors construct infinitely many examples—small Seifert‑fibred, hyperbolic, and toroidal rational homology spheres—showing that the answer is “no”.

The first family of examples comes from Dehn surgery on knots in S³. Using a recent result of Lin, the authors note that the existence of a co‑oriented taut foliation forces a non‑trivial reduced Heegaard‑Floer homology group HF⁺_red(M, s) for some Spinⁿc structure s. Combining this with Heegaard‑Floer obstruction techniques, they prove that if a knot K has genus g and the surgery slope p/q satisfies |p/q| > 2g(K)−1 with even denominator q, then the surgered manifold K(p/q) cannot support a co‑oriented taut foliation with zero Euler class (Theorem 1.3). This improves earlier results that required the foliation to be transverse to the surgery core. The theorem applies to a wide range of knots: fibre‑positive knots, alternating non‑torus knots, and many Montesinos knots (Corollary 1.4). In particular, every even‑denominator large surgery on the figure‑eight knot yields a hyperbolic rational homology sphere that admits a co‑oriented taut foliation (hence a left‑orderable fundamental group) but no such foliation with vanishing Euler class. Similar toroidal examples arise when the knot’s JSJ graph is not a rooted interval (Corollary 1.5).

The second major contribution concerns Seifert‑fibred rational homology spheres. Any co‑oriented taut foliation on such a manifold must be horizontal, so its tangent plane field coincides with the normal bundle ν_M of the Seifert fibration. The authors give a complete arithmetic criterion for ν_M to have trivial Euler class (Theorem 1.9). Specifically, there must exist an integer m satisfying two congruences: m·a_i ≡ 1 (mod b_i) for each exceptional fibre (with Seifert invariants a_i/b_i) and m·e(M) = χ(B), where e(M) is the Seifert Euler number and χ(B) the orbifold Euler characteristic of the base. This condition is both necessary and sufficient. Consequences include a classification of when the Euler class can vanish (Corollaries 1.10, 1.11) and a new proof using discrete faithful PSL(2,ℝ) representations of the base orbifold’s fundamental group when the base is hyperbolic (Section 4). Moreover, the authors show that if the base orbifold is hyperbolic or Euclidean and e(ν_M)=0, then the manifold necessarily admits a horizontal foliation (Theorem 1.12).

Finally, the paper demonstrates that for any finite abelian group G containing an element of order at least three, there are infinitely many Seifert‑fibred rational homology spheres with H₁(M) ≅ G that admit co‑oriented taut foliations but no foliation with zero Euler class (Theorem 1.13). All these manifolds have left‑orderable fundamental groups, confirming the L‑space conjecture in these cases, yet the left orders cannot be obtained from lifts of the universal circle actions associated to the foliations.

Overall, the work combines Heegaard‑Floer homology, knot theory, Seifert‑fibration topology, and the theory of Fuchsian group representations to produce a rich supply of counter‑examples to the “zero Euler class” expectation. It clarifies the limitations of universal‑circle methods for producing left‑orderings and deepens our understanding of the interplay between taut foliations, Euler classes, and the L‑space conjecture.