[Paper Review] A topological grading on bordered Heegaard Floer homology

Yang Huang, Vinicius G. B. Ramos|arXiv (Cornell University)|Nov 30, 2012

Geometric and Algebraic Topology6 references3 citations

TL;DR

This paper introduces a canonical topological grading on bordered Heegaard Floer homology using homotopy classes of nonvanishing vector fields on 3-manifolds with boundary. It generalizes the absolute grading from closed 3-manifolds to bordered settings by defining a groupoid of co-oriented plane fields on $F \times [0,1]$, and proves that the gluing map preserves this grading, extending the absolute grading to the bordered context.

ABSTRACT

In this paper, we construct a canonical grading on bordered Heegaard Floer homology by homotopy classes of nonvanishing vector fields. This grading is a generalization of our construction of an absolute grading on Heegaard Floer homology and it extends the well-known grading with values in a noncommutative group defined by Lipshitz-Ozsváth-Thurston.

Motivation & Objective

• To extend the absolute grading on Heegaard Floer homology from closed 3-manifolds to bordered 3-manifolds with boundary.
• To define a geometric grading on the bordered invariants $\widehat{\mathit{CFA}}(Y)$ and $\widehat{\mathit{CFD}}(Y)$ using homotopy classes of nonvanishing vector fields.
• To construct a grading groupoid $G(F)$ for the strand algebra $\mathcal{A}(F)$ associated to a parametrized surface $F$.
• To prove that the bordered gluing map $\Phi$ preserves the grading, linking the relative grading on the summands to the absolute grading on the closed manifold.

Proposed method

• Define a groupoid $G(F)$ of co-oriented plane fields on $F \times [0,1]$ modulo homotopy, with a $\mathbb{Z}$-action $\lambda^n$.
• Construct a grading function $\mathrm{gr}$ on the strand algebra $\mathcal{A}(F)$ with values in $G(F)$, satisfying $\mathrm{gr}(a \cdot b) = \mathrm{gr}(a) \cdot \mathrm{gr}(b)$ and $\mathrm{gr}(\partial a) = \lambda^{-1} \mathrm{gr}(a)$.
• Define a bimodule $S(Y)$ with left $G(-F)$ and right $G(F)$ actions, serving as the grading set for $\widehat{\mathit{CFA}}(Y)$ and $\widehat{\mathit{CFD}}(Y)$.
• Establish a gluing map $\Psi: S(Y_1) \otimes_{G(F)} S(Y_2) \to \mathrm{Vect}(Y)$ that maps the tensor product of gradings to the absolute grading on the closed manifold $Y = Y_1 \cup_F Y_2$.
• Use the Pontryagin-Thom construction to relate homotopy classes of vector fields on $Y_1$ and $Y_2$ to those on the glued manifold $Y$.
• Prove that the gluing map $\Phi$ is grading-preserving, i.e., $\widetilde{\mathrm{gr}}(\Phi(a \otimes b)) = \Psi(\mathrm{gr}(a) \otimes \mathrm{gr}(b))$.

Experimental results

Research questions

• RQ1 Can the absolute grading on Heegaard Floer homology for closed 3-manifolds be extended to bordered 3-manifolds with boundary?
• RQ2 How can a geometric grading on the bordered invariants $\widehat{\mathit{CFA}}(Y)$ and $\widehat{\mathit{CFD}}(Y)$ be defined using topological data such as vector fields?
• RQ3 Is the gluing map in bordered Heegaard Floer homology compatible with this new grading structure?
• RQ4 What is the algebraic structure of the grading set, and how does it interact with the $\mathcal{A}_\infty$-module actions?
• RQ5 Can the grading on the bordered invariants be related to the absolute grading on the closed manifold obtained by gluing?

Key findings

• A canonical grading on the bordered Heegaard Floer homology invariants $\widehat{\mathit{CFA}}(Y)$ and $\widehat{\mathit{CFD}}(Y)$ is constructed using homotopy classes of nonvanishing vector fields on $Y$.
• The grading takes values in a groupoid $G(F)$ of co-oriented plane fields on $F \times [0,1]$ modulo homotopy, equipped with a $\mathbb{Z}$-action $\lambda^n$.
• The grading satisfies $\mathrm{gr}(a \cdot b) = \mathrm{gr}(a) \cdot \mathrm{gr}(b)$ and $\mathrm{gr}(\partial a) = \lambda^{-1} \mathrm{gr}(a)$ for generators of $\mathcal{A}(F)$.
• For $\widehat{\mathit{CFA}}(Y)$, the grading of a module action satisfies $\mathrm{gr}(m_{l+1}(x; a_1, \dots, a_l)) = \lambda^{l-1} \mathrm{gr}(x) \cdot \mathrm{gr}(a_1) \cdots \mathrm{gr}(a_l)$.
• The gluing map $\Phi: \widehat{\mathit{CFA}}(Y_1) \widetilde{\otimes} \widehat{\mathit{CFD}}(Y_2) \to \widehat{\mathit{CF}}(Y_1 \cup_F Y_2)$ preserves the grading, with $\widetilde{\mathrm{gr}}(\Phi(a \otimes b)) = \Psi(\mathrm{gr}(a) \otimes \mathrm{gr}(b))$.
• The map $\Psi: S(Y_1) \otimes_{G(F)} S(Y_2) \to \mathrm{Vect}(Y)$ is a bijection, establishing a one-to-one correspondence between compatible graded structures and absolute vector field homotopy classes on the closed manifold.