Reduced Heegaard Floer homology ($H{F}_{\mathrm{red}}^{\pm }$)

The reduced minus and plus groups areThe connecting homomorphism of the fundamental long exact sequence induces an isomorphism $\delta :H{F}_{\mathrm{red}}^{+}\to H{F}_{\mathrm{red}}^{-}$ 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 $U$.