A Floer homology invariant for $3$-orbifolds via bordered Floer theory

TL;DR

This paper introduces a new Floer homology invariant for 3-orbifolds with knot singularities, extending Heegaard Floer theory to orbifolds and relating it to classical invariants via bordered Floer techniques.

Contribution

It constructs the invariant O for 3-orbifolds using bordered Floer theory, generalizing the hat flavor of Heegaard Floer homology and linking it to the orbifold's homology and Dehn surgery.

Findings

O categorifies the orbifold's first homology order.

O behaves like  in many cases.

Relationship between O and classical  is determined by the -invariant .

Abstract

Using bordered Floer theory, we construct an invariant $\widehat{\mathit{HFO}}(Y^{\text{orb}})$ for $3$-orbifolds $Y^{\text{orb}}$ with singular set a knot that generalizes the hat flavor $\widehat{\mathit{HF}}(Y)$ of Heegaard Floer homology for closed $3$-manifolds $Y$. We show that for a large class of $3$-orbifolds, $\widehat{\mathit{HFO}}$ behaves like $\widehat{\mathit{HF}}$ in that $\widehat{\mathit{HFO}}$, together with a relative $\mathbb{Z}_2$-grading, categorifies the order of $H_1^{\text{orb}}$. When $Y^{\text{orb}}$ arises as Dehn surgery on an integer-framed knot in $S^3$, we use the $\{-1,0,1\}$-valued knot invariant $\varepsilon$ to determine the relationship between $\widehat{\mathit{HFO}}(Y^{\text{orb}})$ and $\widehat{\mathit{HF}}(Y)$ of the $3$-manifold $Y$ underlying $Y^{\text{orb}}$.