L-spaces and taut foliations on 3-manifolds

The thesis mainly focuses on the “L-space conjecture”. This conjecture is a driving force in modern research within low-dimensional topology and it predicts a way to organise closed, connected, oriented 3-manifolds according to their “complexity”, which can be measured in three, conjecturally equivalent, ways. The three properties that are involved in the conjecture belong to areas of low-dimensional topology that, at least apparently, seem to have very little in common.More precisely the conjecture states that for an irreducible and orientable rational homology sphere M the following facts are equivalent:1)     M is not an L-space;2)     M supports a coorientable taut foliation;3)     the fundamental group of M is left-orderable. This conjecture, that was proposed by Boyer-Gordon-Watson and Juhász, has now been proved for 3-manifolds that can be decomposed along essential tori into non-hyperbolic pieces. Moreover, it has been proved by Ozsváth and Szabó that if a manifold contains a coorientable taut foliation then it is not an L-space. In the thesis we study manifolds that can be obtained as Dehn surgery on fibered hyperbolic two-bridge links with two components. This is an infinite family of hyperbolic links that contains the Whitehead link. The main result is that for these manifolds the equivalence between items 1) and 2) in the conjecture holds. Moreover for every fibered hyperbolic two-bridge link L we are able to determine exactly the set of L-space surgery slopes. On the other hand, the construction of the taut foliations is fairly explicit and we are able to exploit it in two different directions. In one direction, we deduce that some types of satellite knots (that generalise Whitehead doubles) have coorientable taut foliation on all non-meridional surgeries. In the other direction we compute, at least for the surgeries on the Whitehead link, when the Euler classes of these foliations vanish, this implying the left-orderability of the fundamental group of the underlying manifold. The last chapter of the thesis focuses on a slightly different topic, and is a joint work with Ludovico Battista and Leonardo Ferrari. We study and classify the L-spaces among some hyperbolic 3-manifolds with particularly nice geometric properties. This allows us to construct some explicit examples of hyperbolic 4-manifolds with vanishing Seiberg-Witten invariants, addressing a conjecture by LeBrun and answering a question posed by Agol and Lin.