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.

Version: v3 · 2026-09-19 · History
Domain-specific #: 2132
Domain group: Formal Sciences
Origin domain: Mathematics
Subdomains: Symplectic Geometry, Hamiltonian Reduction and Equivariant Cohomology → Mathematics

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.

How would you explain it like I'm…

No faithful explanation at this level. At this level the Kirwan map becomes 'cutting out a piece of a shape and squashing it'. That is exactly the arbitrary restriction or quotient projection the concept is not. The picture also drops the cohomology, the equivariant-to-quotient identification and the regularity hypotheses.

Big Shape to Small Shape Recipe

Some mathematicians study shapes that are being moved around by a group of symmetries, along with a special measuring rule called a moment map. From such a shape you can build a smaller shape in two steps: keep only the points where the measurement reads exactly zero, then treat any points the symmetries move into each other as one single point. The Kirwan map is a fixed recipe that takes symmetry-respecting information about the big shape and turns it into information about the smaller shape. It only works when the zero level is 'nice' and the symmetries behave well there.

Reduction Cohomology Map

In symplectic geometry, a group G can act on a space M in a way described by a moment map μ, and the 'reduced space' is built by taking the level set where μ equals zero and dividing out by G. The Kirwan map is a homomorphism from the equivariant cohomology of M, which tracks topology while respecting the symmetry, to the ordinary cohomology of the reduced space. It works in two steps: restrict classes to the zero level set, then identify equivariant cohomology there with the cohomology of the quotient, which is valid when G acts freely or nearly so and 0 is a regular value. It is not just any restriction map or quotient projection. Whether the map is onto, the famous Kirwan surjectivity theorem, is a separate result proved with Morse theory, and it only holds under its stated hypotheses.

Let M be a Hamiltonian G-space with moment map μ, and suppose 0 is a regular value. The Kirwan map κ: H_G^*(M) → H^*(μ^{-1}(0)/G) is a homomorphism of graded rings. It is defined by first restricting equivariant classes to the invariant level set μ^{-1}(0). Then, if G acts freely there (or with finite stabilizers, in the orbifold setting with suitable coefficients), one uses the isomorphism H_G^*(μ^{-1}(0)) ≅ H^*(μ^{-1}(0)/G). The resulting map is canonical for Hamiltonian reduction; it is not just any restriction map or quotient projection. Kirwan's surjectivity theorem comes from Morse theory applied to the function |μ|², but it is a theorem about the map, not its definition. The construction breaks down at singular levels, when equivariant and ordinary cohomology are conflated, or when coefficients that matter are ignored. Its uses include pushing characteristic classes to symplectic quotients, computing quotient cohomology through kernels and presentations, and linking reduction to localization.

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.
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) — outgoing typed relations

Kirwan map is a kind of Function (Mapping) Prime

The Kirwan map literally assigns each ambient equivariant cohomology class a class on the reduced space through a specified composite homomorphism; Hamiltonian reduction and its hypotheses supply the specialization.

Hierarchy path (1) — routes to 1 parentless root

Kirwan map → is a kind of Function (Mapping)

Neighborhood in Abstraction Space¶

Kirwan map sits in a crowded region of the domain-specific corpus (31^st percentile for distinctiveness): several abstractions share nearly its structure, so a description that fits it tends to fit its neighbors too.

Family — Manifolds, Stacks & Classifying Spaces (18 abstractions)

Nearest neighbors

Computed from structural-signature embeddings · 2026-10-25