Bordered Heegaard Floer homology is a three-manifold invariant which associates to a surface
an algebra
and to a three-manifold
with boundary
identified with
a module over
.
In this paper, we establish naturality properties of this invariant. Changing the diffeomorphism
between
and the
boundary of
tensors the bordered invariant with a suitable bimodule over
. These
bimodules give an action of a suitably based mapping class group on the category of modules
over
.
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
.
We also prove a duality theorem relating the two versions of the
–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.
Keywords
Floer homology, $3$–manifolds, Heegaard diagrams, mapping
class group, Hochschild homology