Kernel, cokernel, and a top-degree obstruction for contact Kirwan maps

Definitions and mathematical statement

The owned source is E. Lerman's Question $3.1$ and C. Boyer's Comment $3.2$, concerning the kernel, cokernel and possible restricted surjectivity of a contact Kirwan map. For a contact action preserving a contact form $\alpha$, its circle moment map is $\mu=\alpha(\xi_M)$. At a regular zero level the ordinary equivariant restriction is \[ \kappa:H_{S^1}^*(M;A)\longrightarrow H_{S^1}^*(\mu^{-1}(0);A), \] identified with quotient cohomology when the action is free. The aid uses coefficients $A={\mathbb{Z}},{\mathbb{Q}}$, or ${\mathbb{R}}$.

Lerman's example is the unit $S^3\subset{\mathbb{C}}^2$, with standard contact form and free action \[ \lambda\cdot(z_1,z_2)=(\lambda z_1,\lambda^{-1}z_2),\qquad \mu=|z_1|^2-|z_2|^2. \] The full quotient is $S^2\cong\mathbb{CP}^1$, while the zero-level quotient is $S^1$, identified with its equator. Consequently $\kappa$ is equator restriction \[ A[u]/(u^2)=H^*(S^2;A)\longrightarrow H^*(S^1;A),\qquad |u|=2, \] with \[ \ker\kappa=(u),\qquad \operatorname{coker}\kappa\cong A \text{ concentrated in degree }1. \] Cokernel here means the graded $A$-module cokernel, not a quotient ring by an ideal generated by the image.

The aid's broader obstruction assumes a compact connected cooriented contact manifold $M^{2n+1}$, $n\geq1$, a free coorientation-preserving contact circle action, an invariant contact form, and a nonempty regular zero level. Then $B=M/S^1$ is an oriented compact $2n$-manifold, $\mu$ descends to a function on $B$, and $Y=\mu^{-1}(0)/S^1$ is a separating oriented hypersurface. The map is $i^*:H^*(B;A)\to H^*(Y;A)$. The pair sequence gives, degreewise, \[ \ker i^*=\operatorname{im}(H^k(B,Y;A)\to H^k(B;A)), \] \[ \operatorname{coker}i^*\cong \ker(H^{k+1}(B,Y;A)\to H^{k+1}(B;A)). \] Orient the components of $Y$ as the boundary of one side. Integration-sum on their top cohomology annihilates restriction from $B$ and induces a surjection \[ \operatorname{coker}\left(H^{2n-1}(B;A)\to H^{2n-1}(Y;A)\right) \longrightarrow A. \] Thus this ordinary map is never fully surjective under these hypotheses. If $Y$ is connected, restriction in that degree is zero and the cokernel is exactly $A$. Freeness, regularity, compactness, coorientation and nonemptiness are essential parts of the reported theorem; no conclusion for all nonfree contact actions is asserted.

Boyer asks which contact or $3$-Sasakian classes might nevertheless admit surjectivity, possibly only up to a middle dimension as in the preceding conjecture. That earlier conjecture and its precise degree range are absent, so no numerical range is invented.

The extraction overcaptures the beginning of Section $4$, which the supplied verification assigns to later records. For provenance only, its Goldin--Holm--Knutson theorem concerns a compact Hamiltonian torus space, regular moment value, a ring on $\bigoplus_{g\in T}H_T^*(M^g;{\mathbb{Q}})$, and a natural surjection to Chen--Ruan orbifold cohomology of the symplectic quotient. Its following Lerman comment distinguishes Haefliger's earlier orbifold cohomology notion. These are neighboring statements, not a contact-surjectivity theorem or part of the owned question.

Short English statement

For Lerman's contact three-sphere example, the map is equator restriction: its kernel is the sphere's degree-two class and its module cokernel is one degree-one copy of the coefficients. The aid extends the failure to free compact contact circle actions with nonempty regular zero quotient, using its separating-hypersurface top cohomology. Restricted and $3$-Sasakian variants remain questions.

Sources

• Local UnsolvedMath registry, permanent ID $20001954$, source code catalogue entry; canonical collection aim-geometry-notes.json, notes[291].
• AIM workshop "Moment maps and surjectivity in various geometries," page $4$, Question $3.1$ (E. Lerman) and Comment $3.2$ (C. Boyer): https://aimath.org/WWN/momentmaps/momentmaps.pdf. The supplied verification excludes overcaptured Section $4$, Theorem $4.1$, and Comment $4.2$ from this owned item. No PDF was opened in this session.
• Neighboring attributions preserved: Goldin--Holm--Knutson, Chen--Ruan orbifold cohomology and Lerman's Haefliger comment. No full bibliographic entries are supplied for these or the contact-map assertions. Original remarks and literature are empty, with no proposer-field value or year.
• Supplied research attempt $1$, partial_result, run status partially_solved, rigor label "proved," novelty confidence "low." Registry classification PARTIAL-PROGRESS, updated 14 August 2026.