Bordered Heegaard Floer homology is an invariant for 3-manifolds, which associates to a surface F an algebra A(Z), and to a 3-manifold Y with boundary, together with an orientation-preserving diffeomorphism φ: F → ∂Y, a module over A(Z). In [10] we defined relative Z/2 differential gradings on the algebra A(Z) and the modules over it. In this paper, we turn the relative grading into an absolute one, and show that the resulting Z/2-graded module is an invariant of the bordered 3-manifold.

I am grateful to Robert Lipshitz for many inspiring conversations, and for his valuable comments on earlier drafts of this paper. I am also thankful to Paolo Ghiggini, Jonathan Hanselman, and Eamonn Tweedy for useful discussions, and to the referee for helpful comments and corrections. A large part of this work was completed during an informal visit at UQAM in Summer 2013; I thank Steve Boyer and Olivier Collin for their hospitality.