= Variants of Heegaard Floer homology
{title2=$HF^-,HF^\infty,HF^+,\widehat{HF}$}
Use the convention that $CF^-$ is generated by $[\mathbf x,i]$ with $i<0$, and $U[\mathbf x,i]=[\mathbf x,i-1]$. The <chain complexes> are
$$
CF^\infty=CF^-\otimes_{\mathbb Z[U]}\mathbb Z[U,U^{-1}],\qquad CF^+=CF^\infty/CF^-,\qquad \widehat{CF}=\ker(U:CF^+\to CF^+).
$$
The last complex is generated by $[\mathbf x,0]$. Equivalently it is $CF^-/UCF^-$ with its absolute grading shifted up by two. The short exact sequence $0\to CF^-\to CF^\infty\to CF^+\to0$ gives the fundamental <long exact sequence>. For a rational homology sphere in each <Spin-c structure>, $HF^\infty\cong\mathbb Z[U,U^{-1}]$. 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 $d-2$.
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