Bordered invariants in low-dimensional topology.

Abstract

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.