In this thesis we present two projects. In the first project, which covers Chapters 2 and 3, we construct an algebraic version of Lagrangian Floer homology for immersed curves in a surface with boundary. We first associate to the surface an algebra A. Then to an immersed curve L inside the surface we associate an A∞ module M(L) over A. Then we prove that Lagrangian Floer homology HF∗(L, L') is isomorphic to a suitable algebraic pairing of modules M(L) and M(L'). We apply this theory to the pillowcase homology construction; namely we enhance it by extending the construction from knots to tangles: given a 4-ended tangle inside a 3-ball, after associating to it an immersed unobstructed curve inside the pillowcase, one can further associate an A∞ module to that curve.

In the second project, which is described in Chapter 4, we compare two different types of mapping class invariants: the Hochschild homology of A∞ bimodules coming from bordered Heegaard Floer homology, and fixed point Floer cohomology. We first develop effective methods to compute bimodule invariants and their Hochschild homology in the genus two case. We then compare the resulting computations to fixed point Floer cohomology, and make a conjecture that the two invariants are isomorphic. We also discuss a construction of a map potentially giving the isomorphism. It comes as an open-closed map in the context of a surface being viewed as a 0-dimensional Lefschetz fibration over C.

A common method of constructing 3-manifolds is via Dehn surgery on knots. A pair of surgeries on a knot is called purely cosmetic if the pair of resulting 3-manifolds are homeomorphic as oriented manifolds, whereas it is said to be chirally cosmetic if they result in homeomorphic manifolds with opposite orientations. An outstanding conjecture predicts that no nontrivial knots admit any purely cosmetic surgeries. We apply certain obstructions from Heegaard Floer homology to show that (nontrivial) knots which arise as the closure of a 3-stranded braid do not admit any purely cosmetic surgeries. Furthermore, we find new obstructions to the existence of chirally cosmetic surgeries coming from Heegaard Floer homology; in particular, we make use of immersed curve formulations of knot Floer homology and the corresponding surgery formula. Combining these with other obstructions involving finite type invariants, we completely classify chirally cosmetic surgeries on odd alternating pretzel knots, and we rule out such surgeries for a large class of Whitehead doubles. Moreover, we rule out cosmetic surgeries for L-space knots along slopes with opposite signs.

In 1976, Thurston proved that taut foliations on closed hyperbolic 3-manifolds have Euler class of norm at most one, and conjectured that, conversely, any Euler class with norm equal to one is Euler class of a taut foliation. We construct the first counterexamples to this conjecture, infinitely many indeed. The counterexamples are constructed by Dehn surgeries on certain fibered hyperbolic 3-manifolds. Moreover, they are constructive in the sense that the monodromy of the fibration map is given in terms of Dehn twists and the surgery coefficient is specified. We also suggest an alternative conjecture in terms of faithful representations of the fundamental group of the 3-manifold into certain group of homeomorphisms.