${{\mathrm{Spin}}}^c(M)$. The tensor product with complex line bundles defines a free and transitive action $H^2(M;{{\mathbb Z}})\times{{\mathrm{Spin}}}^c(M)\to{{\mathrm{Spin}}}^c(M):
(e,\Gamma)\mapsto\Gamma_e$. Each $\Gamma\in{{\mathrm{Spin}}}^c(M)$ has a first Chern class $c_1(\Gamma)\in H^2(M;{{\mathbb Z}})$ and $c_1(\Gamma_e)=c_1(\Gamma)+2e$.

[**2.**]{} Every nondegenerate $2$-form $\rho\in{{\Omega}}^2(M)$, compatible with the orientation, determines a canonical equivalence class of spin$^c$ structures $\Gamma_\rho\in{{\mathrm{Spin}}}^c(M)$, associated to the $\rho$-compatible almost complex structures. It depends only on the homotopy class of $\rho$ and has the first Chern class $c_1(\Gamma_\rho)=c_1(\rho)$.

[**3.**]{} Every nondegenerate $2$-form $\rho\in{{\Omega}}^2(M)$, compatible with the orientation, determines a [*homological orientation of $M$*]{}, i.e.  an orientation ${{\frak o}}_\rho$ of the real vector space $H^0(M;{{\mathbb R}})\oplus H^1(M;{{\mathbb R}})\oplus H^{2,+}(M;{{\mathbb R}})$ (see Donaldson [@D1]).

[**4.**]{} If $b^+=\dim H^{2,+}(M;{{\mathbb R}})=1$ then the [*positive cone*]{} $${{\mathcal K}}:=\left\{a\in H^2(M;{{\mathbb R}})\,|\,a^2>0\right\}$$ has two connected components. If $\rho$ is a symplectic form, compatible with the orientation, let $\kappa_\rho\subset{{\mathcal K}}$ be the connected component containing $[\rho]$.

[**5.**]{} The Seiberg–Witten invariants of $M$ take the form of a map $${{\mathrm{Spin}}}^c(M)\to{{\mathbb Z}}:\Gamma\mapsto{{\mathrm{SW}}}(M,{{\frak o}},\Gamma).$$ This map depends on the choice of a homological orientation ${{\frak o}}$ of $M$ (see 3). Changing the homological orientation reverses the sign of the invariant. When $b^+=1$ the Seiberg–Witten invariant also depends on the choice of a connected component $\kappa$ of ${{\mathcal K}}$ (see 