A Refinement in \v{C}ech Cohomology of Coron's Necessary Condition

Coron established a homological obstruction to continuous feedback stabilization of nonlinear control systems $\dot{x}=f(x,u)$ with $f \in C(\Omega,\mathbb{R}^n)$ and $f(0,0)=0$, showing that local asymptotic stabilizability implies the induced homomorphism $f_*$ satisfies $f_*\big(H_{n-1}(\Sigma_\epsilon)\big)=H_{n-1}(S^{n-1})$, where $\Sigma_\epsilon:=\Big(\big(\mathbb{B}_\epsilon^{\mathbb{R}^n}(0)\times\mathbb{B}_\epsilon^{\mathbb{R}^m}(0)\big)\cap \Omega\Big)\setminus f^{-1}(0)$. In this paper, we refine Coron's necessary condition using \v{C}ech cohomology and the Vietoris-Begle mapping theorem. Specifically, we prove that the closed version of $\Sigma_\epsilon$ must be a \v{C}ech cohomology $(n-1)$-sphere and that the restriction of $f$ to this subset induces an isomorphism on its \v{C}ech cohomology groups in all degrees. This strengthens Coron's condition from a constraint on the top class to a full cohomological rigidity statement.