Combinatorial methods in bordered Heegaard Floer homology

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(Iz), the type AA invariant associated to the identity diffeomorphism. This leads to a combinatorial proof of the quasi-invertibility of CFDD(I z).;In the second part, we use the newly constructed CFAA( Iz) 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.