Cohomology targets and sectorwise Kirwan surjectivity

Definitions and mathematical statement

Source comment and question. The aid identifies this extract as E. Lerman's Comment 10.4 referring to absent Question 10.3. It asks which cohomology theory should be used for a reduced space, specifically Chen--Ruan orbifold cohomology or Haefliger cohomology, and suggests seeking an optimal condition on critical sets of the squared moment-map norm. The underlying action, precise reduction, coefficients and original surjectivity map are not included. The final heading "11 Localization formulas for non-compact groups" starts a different section.

For a quotient orbifold $[Z/G]$, Haefliger cohomology means cohomology of its classifying space, modeled here by $H_G^*(Z)$. Chen--Ruan cohomology instead uses inertia sectors, age shifts and an obstruction-bundle product. These targets need not have the same additive dimension. The source therefore cannot be formalized as a single theory-independent surjectivity assertion.

Supplied rational torus framework. For suitable Hamiltonian torus reductions, the aid invokes Goldin--Holm--Knutson: one replaces ordinary equivariant cohomology by inertial equivariant cohomology, with fixed-locus sectors and their product, to obtain the orbifold Kirwan ring surjection over ${\mathbb{Q}}$. A precise general hypothesis list is not supplied in this comment. The following noncompact linear example has a specified proper moment map and is a separate worked family, not an unqualified extension of a compact-manifold theorem.

Weight-$m$ family. Let $m\ge1$, let $S^1$ act on ${\mathbb{C}}$ by $t\cdot z=t^m z$, and choose the moment-map convention \[ \mu(z)=\frac m2|z|^2-a,\qquad a>0. \] Then $Z=\mu^{-1}(0)$ is a circle, with stabilizer $\mu_m=\{t\in S^1:t^m=1\}$, and the quotient stack is $X=[*/\mu_m]$. Stabilizer data are retained: its effective coarse quotient is just a point and is a different target. Rationally, \[ H^*_{\mathrm{Haef}}(X;{\mathbb{Q}})=H^*(B\mu_m;{\mathbb{Q}})={\mathbb{Q}}, \qquad H^*_{\mathrm{CR}}(X;{\mathbb{Q}})\cong{\mathbb{Q}}[\mu_m] =\bigoplus_{g\in\mu_m}{\mathbb{Q}} e_g. \] All reduced sectors have age zero, and $e_ge_h=e_{gh}$. Ordinary restriction from $H^*_{S^1}({\mathbb{C}};{\mathbb{Q}})={\mathbb{Q}}[u]$, $|u|=2$, is onto the Haefliger target; after inclusion as the identity inertia sector its image is only ${\mathbb{Q}} e_1$, so it does not surject onto the Chen--Ruan target when $m>1$. The target-relevant inertial source is \[ \bigoplus_{g\in\mu_m}H^*_{S^1}({\mathbb{C}}^g;{\mathbb{Q}}) =\bigoplus_{g\in\mu_m}{\mathbb{Q}}[u], \] and the aid asserts that its sectorwise restriction reaches every $e_g$, giving the full rational orbifold ring in this family.

Coefficient-sensitive obstruction. For $\|\mu\|^2$, the origin is the negative normal critical locus of weight $m$, with equivariant Euler class $m u$ under the complex orientation. Every relevant $g\in\mu_m$ fixes all of ${\mathbb{C}}$, so the same restriction occurs in each sector. Over a field $\mathbb k$ of characteristic $p>0$ dividing $m$, this Euler class vanishes and \[ H^1(B\mu_m;\mathbb k)\ne0, \qquad H^*_{S^1}({\mathbb{C}};\mathbb k)=\mathbb k[u] \] has no odd degree. Hence each sector restriction fails to be onto. If $p$ does not divide $m$, positive-degree cohomology of $B\mu_m$ vanishes and the sector restrictions are onto. These are statements about classifying-space sector cohomology and restriction maps; no unprovided modular Chen--Ruan product is asserted. This family demonstrates the effect of stabilizer order and coefficients but does not supply the source's requested optimal general critical-set criterion.

Short English statement

The comment stresses that surjectivity depends on the cohomology target. In the supplied weighted-circle example, ordinary rational equivariant cohomology reaches only the identity Chen--Ruan sector, whereas inertial cohomology reaches them all. Characteristics dividing the stabilizer order obstruct even sectorwise surjectivity. No general optimal Morse-theoretic condition is provided.

Sources

• Local UnsolvedMath registry, permanent ID 20001978, source code catalogue entry; canonical aim-geometry-notes.json, notes[315].
• AIM Moment maps and surjectivity in various geometries, extract identified by the aid as E. Lerman's Comment 10.4 referring to Question 10.3: https://aimath.org/WWN/momentmaps/momentmaps.pdf.
• Goldin--Holm--Knutson theorem named in the supplied research summary; no full bibliographic citation or theorem hypotheses are supplied in this record. Chen--Ruan and Haefliger theories are named in the original comment.
• Registry research attempt 1, dated 2026-08-14: partial result, rigor "proved", novelty confidence low. The machine-generated, non-peer-reviewed aid requires independent verification before citation.