= Blowup formula for the mixed Heegaard Floer invariant
{c}
Let $\widetilde M=M\#\overline{\mathbb {CP}}^{,2}$, and let $E$ be the exceptional sphere. Extend a <Spin-c structure> by $c_1(\widetilde{\mathfrak s})=c_1(\mathfrak s)\pm\operatorname{PD}[E]$. Then $\Phi_{\widetilde M,\widetilde{\mathfrak s}}=\Phi_{M,\mathfrak s}$. More generally, an evaluation $\langle c_1(\widetilde{\mathfrak s}),E\rangle=\pm(2\ell+1)$ 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 $\pm1$ case the exponent is zero, so nonvanishing is preserved.
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