The unchanged-product obstruction

Let ${\mathcal{S}}_g$ consist of supported laboratory tensors satisfying $G^{(1)}(h)=0$, and let ${\mathcal{S}}_{\varphi }$ be the solutions of $P\varphi =0$ with smooth data supported in ${\overline{B}}_{r_{\ast }}$. Let

This is a linear subspace of ${\mathcal{S}}_g$. Define the unchanged product map

No topological quotient theorem for this row is needed.