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

Abstract

Using bordered Floer theory, we construct an invariant $\widehat{HFO}(Y^{\text{orb}})$ for $3$-orbifolds $Y^{\text{orb}}$ with singular set a knot that generalizes the hat flavor $\widehat{HF}(Y)$ of Heegaard Floer homology for closed $3$-manifolds $Y$. We show that for a large class of $3$-orbifolds, $\widehat{HFO}$ behaves like $\widehat{HF}$ in that $\widehat{HFO}$, 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 $\epsilon$ to determine the relationship between $\widehat{HFO}(Y^{\text{orb}})$ and $\widehat{HF}(Y)$ of the $3$-manifold $Y$ underlying $Y^{\text{orb}}$.

28 pages, 16 figures, comments welcome!