Combinatorial Methods in Bordered Heegaard Floer Homology

Abstract

In this thesis we give several combinatorial constructions and proofs in bordered Heegaard Floer homology. In the first part, we give an explicit description of a rank-1 model of CFAA(I), the type AA invariant associated to the identity diffeomorphism. This leads to a combinatorial proof of the quasi-invertibility of CFDD(I). In the second part, we use the newly constructed CFAA(I) to describe the type DA invariant CFDA(\tau) for any arcslide \tau. Using this, we give a combinatorial construction of HF(Y) for any 3-manifold Y, and prove that it is a 3-manifold invariant. Along the way, we prove combinatorially that bordered Floer theory gives a linear-categorical representation of the (strongly-based) mapping class groupoid. In the third part, we develop the theory of local type DA bimodules and extension by identity, and use it to give another construction of CFDA(\tau) for arcslides, which leads to a faster algorithm to compute HF for an arbitrary 3-manifold.