arXiv:1410.3043v1 [math.DG] 12 Oct 2014 TRANSVERSE GEOMETRY OF FOLIATIONS CALIBRATED BY NON-DEGENERATE CLOSED 2-FORMS DAVID MART ´ INEZ TORRES, ´ ALVARO DEL PINO, AND FRANCISCO PRESAS Abstract. Codimension one foliated manifolds (M, F ) admitting a closed 2- form ω making each leaf symplectic are a natural generalization of 3-dimensional taut foliations. Remarkably, on such closed foliated manifolds (M, F ) there exists a class of 3-dimensional transverse closed submanifolds W on which F induces a taut foliation F W . Our main result says that the foliated subman- ifold (W, F W ) has the same transverse geometry as (M, F ). More precisely, the inclusion induces an essential equivalence between the corresponding ho- lonomy groupoids. The proof of our main result relies on a leafwise Lefschetz hyperplane theorem, which is of independent interest. 1. Introduction and Statement of Main Results Let F be a foliation by surfaces on a closed 3-dimensional manifold W . The foliation F is called taut if for every leaf there exists a loop C through it such that C ⋔ F (C is everywhere transverse to F ). This topological definition is equivalent to the following differential geometric characterization: there exists a closed 2-form inducing an area form on each leaf, see [21]. Taut foliations have no Reeb components. The latter are a source of flexibility in the construction of foliations by surfaces on 3-manifolds. Hence, it is not surprising that the existence of a Reebless foliation F on W has consequences for both the topology of W and the topology of the pair (W, F ). The most well-known topo- logical constraints for a Reebless foliation are related to the fundamental group of W : the universal covering space of W is diffeomorphic to R 3 [17]; also, work of Novikov on vanishing cycles ensures that the fundamental group of any leaf injects into the fundamental group of F , and that every loop C ⋔ F must be non-trivial in homotopy. A 3-dimensional closed manifold with a taut foliation has additional remarkable properties: there exist metrics making each compact region of a leaf a minimal hypersurface inside its relative homology class (topological tautness equals geometric tautness ), and the taut condition can be reformulated in terms of foliation cycles (topological tautness equals homological tautness ). The notion of tautness has a straightforward generalization to codimension one foliations on closed manifolds of arbitrary dimension (M p , F ). One requires the existence through any leaf of a loop C ⋔ F , or, equivalently [20], the existence of a closed p − 1-form whose restriction to every leaf of F is a volume form. It is still true for taut foliations on arbitrary dimension that topological tautness is equivalent to both geometric tautness [19] or homological tautness [21, 10]. Like- wise, the absence of vanishing cycles in the 3-dimensional case generalizes to the absence of exploding plateaus [1]. However, the lack of exploding plateaus carries no homotopical information. This explains why most of the topological aspects of the rich theory of 3-dimensional taut foliations do not extend to higher dimensions. In fact, taut foliations in high dimensions are very flexible objects, as shown by the h–principle proved in [13]. 1