 Mixed Heegaard Floer invariant

For a closed smooth oriented four-manifold $M$ with $b_2^{+}(M)>1$, remove two balls to obtain $W:S^3\to S^3$, and choose an Admissible cut for a Heegaard Floer mixed map. Positivity and naturality show that the minus map of $W_1$ lands in $H{F}_{\mathrm{red}}^{-}$, while the plus map of $W_2$ descends from $H{F}_{\mathrm{red}}^{+}$. ThusThe coefficient of the bottom plus generator in this map, applied to $U^r{\Theta }^{-}\otimes \zeta$, defines ${\Phi }_{M,\mathfrak{s}}(U^r\otimes \zeta )\in \mathbb{Z}$, up to the orientation sign, with $\zeta \in {\Lambda }^{\ast }(H_1(M)/\mathrm{torsion})$. It vanishes unless $2r+\mathrm{deg}\zeta =(c_1(\mathfrak{s}{)}^2-2\chi (M)-3\sigma (M))/4$. The construction is independent of the admissible cut by the cobordism composition and associativity laws. Two positive pieces explain the condition $b_2^{+}>1$.