Formal Neighborhood Equivariant Cohomology and Kirwan Surjectivity at Singular Values
Abstract
Let G be a compact Lie group acting Hamiltonianly on a symplectic manifold M with moment map μ:M→g*. For a singular value a of μ, the symplectic quotient μ−1(a)/G is stratified and the classical Kirwan map is not defined. We introduce a \emph{formal neighborhood equivariant cohomology} cHG*(a) using the slice-groupoid Cartan sheaf and the closed embedding μ−1(a)/G↪ Na/G for a G-invariant neighborhood Na of μ−1(a). We define a formal Kirwan map κformal:HG*(M)⟶ cHG*(a). We prove that it is surjective under a local equivariant homotopy retraction condition on the slice level sets Cx=μx−1(a|gx)⊂ Sx, which is satisfied. We also prove an equivariant tubular neighborhood theorem for singular level sets: under real analytic/proper or G-invariant Whitney stratified assumptions, there exists a G-invariant open neighborhood U of μ−1(a) and a G-equivariant strong deformation retraction U→μ−1(a). When such a neighborhood exists, the formal cohomology is canonically isomorphic to the Borel equivariant cohomology of the level set, and the formal Kirwan map is the classical restriction map. The formal framework therefore yields a Kirwan-type map at the singular value with the surjectivity property. As an application we recover the even-degree surjectivity theorem of Delarue--Ramacher--Schmitt for S1-actions and clarify why odd-degree surjectivity fails in general.
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BibTeX
@misc{huang2026formal,
title = {{Formal Neighborhood Equivariant Cohomology and Kirwan Surjectivity at Singular Values}},
author = {Zhenxi Huang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.35936},
eprint = {2610.35936},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.35936},
}
RIS
TY - UNPB
TI - Formal Neighborhood Equivariant Cohomology and Kirwan Surjectivity at Singular Values
AU - Zhenxi Huang
PY - 2026
DA - 2026/10/29
PB - arXiv
DO - 10.48550/arxiv.2610.35936
UR - https://arxiv.org/abs/2610.35936
AB - Let G be a compact Lie group acting Hamiltonianly on a symplectic manifold M with moment map μ:M→g^*. For a singular value a of μ, the symplectic quotient μ^−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 μ^−1(a)/G↪ N_a/G for a G-invariant neighborhood N_a of μ^−1(a). We define a formal Kirwan map κ_formal:H_G^*(M)⟶ cH_G^*(a). We prove that it is surjective under a local equivariant homotopy retraction condition on the slice level sets C_x=μ_x^−1(a|_g_x)⊂ S_x, which is satisfied. We also prove an equivariant tubular neighborhood theorem for singular level sets: under real analytic/proper or G-invariant Whitney stratified assumptions, there exists a G-invariant open neighborhood U of μ^−1(a) and a G-equivariant strong deformation retraction U→μ^−1(a). When such a neighborhood exists, the formal cohomology is canonically isomorphic to the Borel equivariant cohomology of the level set, and the formal Kirwan map is the classical restriction map. The formal framework therefore yields a Kirwan-type map at the singular value with the surjectivity property. As an application we recover the even-degree surjectivity theorem of Delarue--Ramacher--Schmitt for S^1-actions and clarify why odd-degree surjectivity fails in general.
ER -