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.