[show abstract][hide abstract] ABSTRACT: Bordered Heegaard Floer homology is a three-manifold invariant which
associates to a surface F an algebra A(F) and to a three-manifold Y with
boundary identified with F a module over A(F). In this paper, we establish
naturality properties of this invariant. Changing the diffeomorphism between F
and the boundary of Y tensors the bordered invariant with a suitable bimodule
over A(F). These bimodules give an action of a suitably based mapping class
group on the category of modules over A(F). The Hochschild homology of such a
bimodule is identified with the knot Floer homology of the associated open book
decomposition. In the course of establishing these results, we also calculate
the homology of A(F). We also prove a duality theorem relating the two version
of the 3-manifold invariant. Finally, in the case of a genus one surface, we
calculate the mapping class group action explicitly. This completes the
description of bordered Heegaard Floer homology for knot complements in terms
of the knot Floer homology.
[show abstract][hide abstract] ABSTRACT: We consider a stabilized version of hat Heegaard Floer homology of a
3-manifold Y (i.e. the U=0 variant of Heegaard Floer homology for closed
3-manifolds). We give a combinatorial algorithm for constructing this
invariant, starting from a Heegaard decomposition for Y, and give a
combinatorial proof of its invariance properties.
[show abstract][hide abstract] ABSTRACT: The aim of this article is to introduce invariants of oriented, smooth, closed four-manifolds, built using the Floer homology theories defined in two earlier papers (math.SG/0101206 and math.SG/0105202). This four-dimensional theory also endows the corresponding three-dimensional theories with additional structure: an absolute grading of certain of its Floer homology groups. The cornerstone of these constructions is the study of holomorphic disks in the symmetric products of Riemann surfaces.