• taut foliations
• foliations
• foliazioni
• Euler class
• classe di Eulero

In this thesis, we introduce the tools needed to present a conjecture proposed by Thurston in 1986, and we present the disproof which was recently given by Yazdi in an article published in 2017.

In the first chapter, we present the definition of 1-codimensional foliations on closed manifolds, and we introduce some basic tools to work with them, such as transversals and holonomy. We are most interested in foliations which satisfy the condition of tautness, which means that all the leaves intersect a closed transversal. Actually, tautness can be defined in other equivalent ways, some of which depend on the choice of a Riemannian metric on the manifold.

In the second chapter we study the Thurston norm, which measures the minimum degree of complexity a surface must have in order to represent an element of the homology, and the corresponding dual norm x* on H²(M, ∂M; ℝ). It turns out that x*(e(TF)) ≤ 1, where F is a taut foliation, and e(TF) denotes the Euler class of the tangent bundle to the foliation, which is an element of the second cohomology. In most cases the equality holds, and this led Thurston to conjecture that, when M is closed, irreducible, and atoroidal, every integral element in the unit sphere of H²(M) is the Euler class of some taut foliation F.

In the third chapter we present the counterexample found by Yazdi. We construct the candidate manifold by performing a Dehn surgery on the mapping torus of f, where f is an appropriate self-homeomorphism of a surface of genus g. After computing the second homology and the Thurston norm, we pick a specific unit element and prove that there are no taut foliations with that element as Euler class.