= Blowup formula for the mixed Heegaard Floer invariant
{c}
Let $M$ be a closed oriented <smooth four-manifold> with $b_2^+(M)>1$, and let
$$
\widetilde M=M\#\overline{\mathbb{CP}}^2
$$
be its blowup, with exceptional <sphere> $E$. A <Spin-c structure> extending one on $M$ with exceptional evaluation $\langle c_1(\widetilde{\mathfrak s}),E\rangle=\pm1$ has the same <Mixed Heegaard Floer invariant>, up to the conventional overall sign. In particular, nonvanishing is preserved.
More generally an evaluation $\pm(2\ell+1)$, with $\ell\geq0$, makes the exceptional-piece <Heegaard Floer cobordism maps> on the minus, infinity and plus versions act by multiplication by $U^{\ell(\ell+1)/2}$. Putting the blowup on one side of an admissible cut and using the cobordism composition law supplies the corresponding mixed-invariant formula. This states the factor at the <map> level, where the placement of $U$ is unambiguous. For $\ell=0$ that factor is one.<Holomorphic triangles and invariants for smooth four-manifolds by Peter Ozsváth and Zoltán Szabó>[Theorem3.7, exceptional cobordism blowup formula]{ref}
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