 Variants of Heegaard Floer homology

Use the convention that $C{F}^{-}$ is generated by $[\mathbf{x},i]$ with $i<0$, and $U[\mathbf{x},i]=[\mathbf{x},i-1]$. The chain complexes areThe last complex is generated by $[\mathbf{x},0]$. Equivalently it is $C{F}^{-}/UC{F}^{-}$ with its absolute grading shifted up by two. The short exact sequence $0\to C{F}^{-}\to C{F}^{\infty }\to C{F}^{+}\to 0$ gives the fundamental long exact sequence. For a rational homology sphere in each Spin-c structure, $H{F}^{\infty }\cong \mathbb{Z}[U,U^{-1}]$. The plus tower is ${\mathcal{T}}_d^{+}=\mathbb{Z}[U,U^{-1}]/U\mathbb{Z}[U]$, with bottom degree $d$; its corresponding minus tower is a free $\mathbb{Z}[U]$ module with top degree $d-2$.