Mixed Heegaard Floer invariant (${\Phi }_{M,\mathfrak{s}}$)

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 $b_2^{+}>1$.

If ${\Phi }_{M,\mathfrak{s}}\ne 0$ and a connected closed oriented embedded surface $S$ has positive genus and $S^2\ge 0$, thenBlow up $S^2$ points on $S$ and choose the $\pm 1$ exceptional determinant classes which increase the Chern pairing on its proper transform by $S^2$. The Blowup formula for the mixed Heegaard Floer invariant preserves nonvanishing, while the proper transform has square zero. A cut disjoint from it allows the relevant cobordism map to pass through $H{F}^{\pm }(S\times S^1,\mathfrak{s}{|}_{S\times S^1})$. These groups vanish when the Chern pairing exceeds $2g-2$, so the mixed map would then vanish, a contradiction. Reverse the surface orientation for the absolute value. The positive-genus condition is essential: a small nullhomologous sphere has square zero and zero pairing and would violate the displayed inequality.

Let $\tilde{M}=M\#{\overline{\mathbb{CP}}}^{,2}$, and let $E$ be the exceptional sphere. Extend a Spin-c structure by $c_1(\tilde{\mathfrak{s}})=c_1(\mathfrak{s})\pm \mathrm{PD}[E]$. Then ${\Phi }_{\tilde{M},\tilde{\mathfrak{s}}}={\Phi }_{M,\mathfrak{s}}$. More generally, an evaluation $⟨c_1(\tilde{\mathfrak{s}}),E⟩=\pm (2\ell +1)$ introduces the factor $U^{\ell (\ell +1)/2}$. This follows by putting the blowup on one side of an admissible cut: the punctured exceptional piece is negative definite, its minus and plus tower maps are multiplication by this power of $U$, and the cobordism composition law gives the formula. In the $\pm 1$ case the exponent is zero, so nonvanishing is preserved.