4). Changing the connected component is governed by the wall crossing formula of Li–Liu [@LL].

[**6.**]{} Let $\rho$ be a symplectic form, compatible with the orientation, and $a:=[\rho]$. When $b^+\ge2$, Taubes proved in [@TAUBES1; @TAUBES2] that $$\label{eq:SWT1}
{{\mathrm{SW}}}(M,{{\frak o}}_\rho,\Gamma_\rho)=1,$$ $$\label{eq:SWT2}
{{\mathrm{SW}}}(M,{{\frak o}}_\rho,\Gamma_{\rho,e})\ne0
\qquad\implies\qquad a\cdot e\ge 0,$$ $$\label{eq:SWT3}
{{\mathrm{SW}}}(M,{{\frak o}}_\rho,\Gamma_{\rho,e})\ne0
\quad\mbox{ and }\quad a\cdot e=0
\qquad\implies\qquad e=0$$ for all $e\in H^2(M;{{\mathbb Z}})$. In [@TAUBES3] Taubes proved that every cohomology class $e$ with ${{\mathrm{SW}}}(M,{{\frak o}}_\rho,\Gamma_{\rho,e})\ne0$ is Poincaré dual to a $\rho$-symplectic submanifold of $M$. These results continue to hold when $b^+=1$, with ${{\mathrm{SW}}}(M,{{\frak o}}_\rho,\Gamma_{\rho,e})$ replaced by ${{\mathrm{SW}}}(M,{{\frak o}}_\rho,\kappa_\rho,\Gamma_{\rho,e})$ (see [@TAUBES1; @TAUBES2; @TAUBES3; @TAUBES4; @TAUBES5]).

An immediate corollary of Taubes’ theorems is the uniqueness of the first Chern class for cohomologous symplectic forms in dimension four.

[**Corollary A.**]{} [*Let $M$ be a closed smooth four-manifold. Two cohomologous symplectic forms on $M$ have equivalent spin$^c$ structures and hence have the same first Chern class.*]{}

Let $\rho,\rho'$ be cohomologous symplectic forms on $M$ and $a:=[\rho]=[\rho']$. Choose the orientation such that $a^2>0$ and choose $e$ such that $\Gamma_{\rho'}=\Gamma_{\rho,e}$. Assume first that $b^+\ge2$. By , ${{\mathrm{SW}}}(M,{{\frak o}}_{\rho'},\Gamma_{\rho,e}) = {{\mathrm{SW}}}(M,{{\frak o}}_{\rho'},\Gamma_{\rho'})=1$ and so ${{\mathrm{SW}}}(M,{{\frak o}}_\rho,\Gamma_{\rho,e})\ne0$. Hence $a\cdot e\ge 0$ by . Interchanging the roles of $\rho$ and $\rho'$ 