wall crossing formula in Li–Liu [@LL] asserts, in the simply connected case, that ${{\mathrm{SW}}}(M,{{\frak o}},\kappa,\Gamma)-{{\mathrm{SW}}}(M,{{\frak o}},-\kappa,\Gamma)=\pm1$ whenever $c_1(\Gamma)^2\ge2\chi+3\sigma$. Since ${{\mathrm{SW}}}(M,{{\frak o}}_\rho,\kappa_\rho,\Gamma_\rho)
-{{\mathrm{SW}}}(M,{{\frak o}}_\rho,\kappa_{\rho'},\Gamma_\rho)$ is even, it follows that ${\kappa_{\rho'}=\kappa_\rho}$. Hence ${{\mathrm{SW}}}(M,{{\frak o}}_{\rho'},\kappa_\rho,\Gamma_\rho) 
= 1 = {{\mathrm{SW}}}(M,{{\frak o}}_\rho,\kappa_\rho,\Gamma_\rho)$, and hence ${{\frak o}}_{\rho'} = {{\frak o}}_\rho$. In [@D1 Prop. 3.25] and [@D2 Lemma 6.4] Donaldson proved that there is a free involution on $\pi_0({{\mathcal R}}^c)$, which reverses the homological orientation. Hence any two symplectic forms in ${{\mathcal R}}^c$ belong to the same connected component. This proves (i). Assertion (ii) follows from (i) and Corollary A.

Examples {#sec:EX}
========

\[ex:open\]Let $M$ be a connected noncompact smooth manifold that admits an almost complex structure. Then the h-principle rules. Namely, a theorem of Gromov asserts that, for every deRham cohomology class $a\in H^2(M;{{\mathbb R}})$, the inclusion of the space ${{\mathcal S}}_a$ of all symplectic forms representing the class $a$ into the space of all nondegenerate $2$-forms is a homotopy equivalence. (See [@GROMOV0], [@GROMOV2 page 84], [@EM Theorem 10.2.2], [@MS1 Theorem 7.34].) This implies the existence statement in \[para:existence\] and shows that the uniqueness problem reduces to topological obstruction theory.

For Euclidean space $M={{\mathbb R}}^{2n}$ the answer to question 3 is positive and the answers to 