Formal Neighborhood Equivariant Cohomology and Kirwan Surjectivity at Singular Values

arXiv:2610.35936v1 Announce Type: new Abstract: Let \(G\) be a compact Lie group acting Hamiltonianly on a symplectic manifold \(M\) with moment map \(\mu:M\to\g^*\). For a singular value \(a\) of \(\mu\), the symplectic quotient \(\mu^{-1}(a)/G\) is stratified and the classical Kirwan map is not defined. We introduce a \emph{formal neighborhood equivariant cohomology} \(\cH_G^*(a)\) using the slice-groupoid Cartan sheaf and the closed embedding \(\mu^{-1}(a)/G\hookrightarrow \mu^{-1}(U_a)/G\) for a \(G\)-invariant neighborhood \(U_a\) of \(a\). We define a formal Kirwan map \[ \kappa_{\mathrm{formal}}:H_G^*(M)\longrightarrow \cH_G^*(a). \] We prove that \(\kappa_{\mathrm{formal}}\) is surjective under a local \emph{equivariant homotopy retraction} condition on the slice level sets \[ C_x=\mu_x^{-1}(a|_{\g_x})\subset S_x, \] which is satisfied, for example, when \(a\) is a regular value of every slice moment map, or when \(a=0\) and \(C_x\) is a \(G_x\)-equivariant cone. The weakest condition we identify is the existence, for each \(x\in\mu^{-1}(a)\), of a local quasi-isomorphism \[ \Car(S_x,G_x)\simeq \cC^\bullet_{C_x} \] in a \(G_x\)-equivariant neighborhood of the orbit, without requiring a global tubular neighborhood. As an application we recover, in the formal framework, the even-degree surjectivity theorem of Delarue--Ramacher--Schmitt for \(S^1\)-actions and clarify why odd-degree surjectivity fails in general.

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