= Reduced Heegaard Floer homology
{c}
{title2=$HF^\pm_{\mathrm{red}}$}
The reduced minus and plus groups are
$$
HF^-_{\mathrm{red}}=\ker(HF^-\to HF^\infty),\qquad HF^+_{\mathrm{red}}=\operatorname{coker}(HF^\infty\to HF^+).
$$
The connecting homomorphism of the fundamental <long exact sequence> induces an isomorphism $\delta:HF^+_{\mathrm{red}}\to HF^-_{\mathrm{red}}$ of degree $-1$. This follows immediately from exactness: its image is the indicated kernel, and its kernel is precisely the image quotiented out on the plus side. These groups isolate the information lost by inverting $U$.
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