[Paper Review] An absolute Z/2 grading on bordered Heegaard Floer homology

Ina Petkova|arXiv (Cornell University)|Jan 12, 2014

Geometric and Algebraic Topology10 references3 citations

TL;DR

This paper establishes an absolute Z/2 grading on bordered Heegaard Floer homology by introducing a canonical choice of ordering and orientation for the α- and β-curves in a bordered Heegaard diagram, lifting a previously relative Z/2 grading to an absolute one. The resulting Z/2-graded type D structure is invariant under homotopy equivalence and independent of the choice of admissible almost complex structure or Heegaard diagram, providing a topological invariant of the bordered 3-manifold.

ABSTRACT

Bordered Heegaard Floer homology is an invariant for 3-manifolds, which associates to a surface F an algebra A(F), and to a 3-manifold Y with boundary, together with an orientation-preserving diffeomorphism from F to the boundary of Y, a module over A(F). In a previous paper, we defined relative Z/2 differential gradings on the algebra A(F) 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.

Motivation & Objective

• To resolve the indeterminacy in the relative Z/2 grading of bordered Heegaard Floer homology by defining an absolute grading.
• To construct a canonical, computable Z/2 grading on the type D module CFD̂(H) using a bordered Heegaard diagram H.
• To prove that the resulting Z/2-graded module is invariant under homotopy equivalence and independent of choices of almost complex structures and admissible diagrams.
• To show that the graded module structure, not just its span, is a topological invariant of the bordered 3-manifold.

Proposed method

• Define a canonical total ordering on the set of homology classes of the surface using a matched circle and a preferred ordering of the α-arcs.
• Assign orientations to the α- and β-curves based on a fixed reference Heegaard diagram and the induced ordering on homology classes.
• Construct a Z/2 grading on the generators of the type D module by comparing their position in the ordered set of Lagrangian bases.
• Ensure the grading satisfies the homogeneity condition m(ax) = m(a) + m(x) and the differential condition m(∂x) = m(x) − 1.
• Prove invariance under changes in the almost complex structure and Heegaard diagram by showing the grading structure depends only on the homology data and canonical ordering.
• Use the compatibility of the intersection form with the canonical orientation to show independence from the choice of reference Heegaard diagram.

Experimental results

Research questions

• RQ1 Can a relative Z/2 grading in bordered Heegaard Floer homology be lifted to an absolute grading using a canonical construction?
• RQ2 Is the resulting absolute Z/2 grading independent of the choice of admissible Heegaard diagram and almost complex structure?
• RQ3 Does the absolute grading on the type D module recover the absolute Z/2 grading on the closed 3-manifold invariant after gluing?
• RQ4 Can the grading be defined uniformly across all spin^c structures using only the Heegaard diagram data?
• RQ5 Is the graded module structure itself a topological invariant, not just its span?

Key findings

• An absolute Z/2 grading is defined on the type D module CFD̂(H) of a bordered Heegaard diagram H, using a canonical ordering and orientation of the α- and β-curves.
• The grading satisfies the homogeneity condition m(ax) = m(a) + m(x) and the differential condition m(∂x) = m(x) − 1, making it compatible with the algebraic structure.
• The resulting Z/2-graded module is invariant under homotopy equivalence and independent of the choice of admissible almost complex structure and Heegaard diagram.
• The graded module structure [CFD̂(H)] is a topological invariant of the bordered 3-manifold, eliminating the sign indeterminacy in the span.
• The construction generalizes to bimodules and extends previous results to decategorification with Z coefficients.
• The absolute grading does not recover the absolute Z/2 grading on the closed manifold invariant after gluing, as shown by Hanselman.