Kirwan map¶
Restrict equivariant cohomology classes of a Hamiltonian group space to a regular moment-map level set and identify them with ordinary cohomology classes on the resulting symplectic quotient.
Core Idea¶
For a Hamiltonian \(G\)-space with moment map \(\mu\) and regular value \(0\), the Kirwan map is the graded-ring homomorphism \(\kappa:H_G^*(M)\to H^*(\mu^{-1}(0)/G)\) obtained by restriction to \(\mu^{-1}(0)\) followed by the equivariant-to-quotient identification under the relevant freeness or orbifold hypotheses. An equivariant class on the ambient Hamiltonian space restricts to the invariant level set, and when the group action there has the required regularity its equivariant cohomology represents the cohomology of the reduced space; Morse-theoretic analysis of the moment-map norm supplies the classical surjectivity theorem.
Scope of Application¶
Kirwan map applies when the analyst can specify a Hamiltonian action of a compact Lie group on a symplectic manifold, a moment map, a suitably regular level set, and the associated symplectic quotient and establish that the map has the ambient equivariant cohomology as source, the reduced-space cohomology as target, and is induced by level-set restriction plus the justified quotient identification. The entry centers the classical cohomological construction; singular, noncompact, K-theoretic, stacky, and derived variants require separately stated hypotheses and targets.
Clarity¶
A clear claim names the carrier, governing rule, assumptions, and recognition test. This matters because Kirwan map can be used for related K-theory or generalized-cohomology maps, and prose often merges the map's definition with Kirwan's surjectivity theorem. The disciplined statement is that the object counts as Kirwan map exactly when the map has the ambient equivariant cohomology as source, the reduced-space cohomology as target, and is induced by level-set restriction plus the justified quotient identification
Manages Complexity¶
The abstraction compresses abelian and nonabelian actions, smooth and orbifold quotients, different regular levels, generalized cohomology theories, K-theoretic analogues, and singular-reduction extensions into a stable carrier, rule, invariant, and failure boundary. It makes comparison tractable while retaining the variables that control validity.
Compression can hide assumptions. A responsible use therefore declares compactness, group, coefficient ring, moment-map level, regularity, free versus locally free action, source grading, target singularity, surjectivity, kernel, and localization data and returns to the full diagnostic whenever a convention or boundary case changes.
Abstract Reasoning¶
- Type the carrier. Establish a Hamiltonian action of a compact Lie group on a symplectic manifold, a moment map, a suitably regular level set, and the associated symplectic quotient and reject examples from a different problem. 2. Lock the rule. Express that the map has the ambient equivariant cohomology as source, the reduced-space cohomology as target, and is induced by level-set restriction plus the justified quotient identification independently of one notation or implementation.
Knowledge Transfer¶
Transfer within symplectic geometry is strong when new cases preserve the same carrier, mechanism, and diagnostic. The move from For a compact Hamiltonian \(G\)-manifold with \(0\) a regular value and a free action on \(\mu^{-1}(0)\), restriction followed by \(H_G^*(\mu^{-1}(0))\cong H^*(M/\!/G)\) gives the Kirwan map to the smooth quotient. to In a torus reduction, one can use fixed-point and moment-polytope data to identify classes that vanish after reduction and thereby present the quotient's cohomology ring. demonstrates that continuity.
Relationships to Other Abstractions¶
Current abstraction Kirwan map Domain-specific
Parents (1) — more general patterns this builds on
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Kirwan map is a kind of Function (Mapping) Prime
The proposed strict upward parent is
prime:function_mapping.
Hierarchy path (1) — routes to 1 parentless root
- Kirwan map → Function (Mapping)
Neighborhood in Abstraction Space¶
Kirwan map sits in a moderately populated region (50th percentile for distinctiveness): it has near-neighbors but no dense thicket of look-alikes.
Family — Differential Topology & Geometric Structure (11 abstractions)
Nearest neighbors
- Equivariant differential form — 0.89
- Geometric quotient — 0.88
- Solvmanifold — 0.88
- Symplectization — 0.88
- Quotient stack — 0.88
Computed from structural-signature embeddings · 2026-09-08