Localization formula for equivariant cohomology¶

The localization theorem for equivariant cohomology in non-rational coefficients is discussed in Daniel Quillen's papers.

Version: v1 · 2026-09-28 · History
Domain-specific #: 10461
Domain group: Formal Sciences
Origin domain: Mathematics
Subdomains: Equivariant Cohomology, Algebraic Topology → Mathematics

Core Idea¶

Localization formula for equivariant cohomology is treated here as the recurring mathematics, logic, and statistics identity summarized by this source-grounded definition: The localization theorem for equivariant cohomology in non-rational coefficients is discussed in Daniel Quillen's papers.

In differential geometry, the localization formula states that for an equivariantly closed equivariant differential form \alpha on an orbifold M with a torus action and for a sufficient small \xi in the Lie algebra of the torus T, we have. {1 \over d_M} \int_M \alpha(\xi) = \sum_F {1 \over d_F} \int_F {\alpha(\xi) \over e_T(F)(\xi)}. where the sum runs over all connected components F of the set M^T of fixed points, d_M is the orbifold multiplicity of M (which equals 1 if M is a manifold), and e_T(F) is the equivariant Euler form of the normal bundle of F.

The formula allows one to compute the equivariant cohomology ring of the orbifold M (a particular kind of differentiable stack) from the equivariant cohomology of its fixed point components, up to multiplicities and Euler forms. No analog of such results holds in the non-equivariant cohomology. One important consequence of the formula is the Duistermaat–Heckman theorem, which states: supposing there is a Hamiltonian circle action (for simplicity) on a compact symplectic manifold M of dimension 2n,.

For Localization formula for equivariant cohomology, the abstraction is narrower than the article's general subject matter: a positive case must preserve The localization theorem for equivariant cohomology in non-rational coefficients is discussed in Daniel Quillen's papers. Retaining only the name, a familiar example, or a downstream effect is insufficient. The specialist roles and tests remain anchored in mathematics, logic, and statistics, which is why this identity is domain-specific rather than prime.

Structural Signature¶

Sig role-phrases:

Defining carrier — The formula allows one to compute the equivariant cohomology ring of the orbifold M (a particular kind of differentiable stack) from the equivariant cohomology of its fixed point components, up to multiplicities and Euler forms.
Constitutive relation — where H is Hamiltonian for the circle action, the sum is over points fixed by the circle action and \alpha_j(p) are eigenvalues on the tangent space at p (cf.
Operating condition — An alternative name for the formula is Borel cohomology, after Armand Borel.
Recognition evidence — The localization theorem states that the equivariant cohomology can be recovered, up to torsion elements, from the equivariant cohomology of the fixed point subset.
Admissible variation — But, there is still a version of the localization theorem for non-abelian actions.
Characteristic consequence — This does not extend, in verbatim, to the non-abelian action.
Failure boundary — In differential geometry, the localization formula states that for an equivariantly closed equivariant differential form \alpha on an orbifold M with a torus action and for a sufficient small \xi in the Lie algebra of the torus T, we have.

What It Is Not¶

Not the whole field of mathematics, logic, and statistics. The node requires the specific identity stated by The localization theorem for equivariant cohomology in non-rational coefficients is discussed in Daniel Quillen's papers.
Not an over-broad reading. This does not extend, in verbatim, to the non-abelian action.
Not an over-broad reading. In differential geometry, the localization formula states that for an equivariantly closed equivariant differential form \alpha on an orbifold M with a torus action and for a sufficient small \xi in the Lie algebra of the torus T, we have.
Not an over-broad reading. The formula allows one to compute the equivariant cohomology ring of the orbifold M (a particular kind of differentiable stack) from the equivariant cohomology of its fixed point components, up to multiplicities and Euler forms.
Not automatically Verdier duality. Retrieval proximity does not establish equivalence; the two identities must be compared by carrier, operation, and failure boundary.

Scope of Application¶

Localization formula for equivariant cohomology applies literally inside mathematics, logic, and statistics wherever the source-defined carrier and relation can be established. Its documented habitats include:

Documented setting. The formula allows one to compute the equivariant cohomology ring of the orbifold M (a particular kind of differentiable stack) from the equivariant cohomology of its fixed point components, up to multiplicities and Euler forms.
History. An alternative name for the formula is Borel cohomology, after Armand Borel.
Non-abelian localization. The localization theorem states that the equivariant cohomology can be recovered, up to torsion elements, from the equivariant cohomology of the fixed point subset.
Non-abelian localization. But, there is still a version of the localization theorem for non-abelian actions.
Non-abelian localization. This does not extend, in verbatim, to the non-abelian action.
Documented setting. In differential geometry, the localization formula states that for an equivariantly closed equivariant differential form \alpha on an orbifold M with a torus action and for a sufficient small \xi in the Lie algebra of the torus T, we have.

Outside mathematics, logic, and statistics, the name should be retained only when these same operational conditions survive; otherwise the comparison belongs to the broader parent Representation or should be marked as analogy.

Clarity¶

A clear use of Localization formula for equivariant cohomology names the carrier, the operative relation, and the conditions under which the source treats the identity as present. The minimal definition is The localization theorem for equivariant cohomology in non-rational coefficients is discussed in Daniel Quillen's papers. The strongest recognition evidence in the frozen account is: The localization theorem states that the equivariant cohomology can be recovered, up to torsion elements, from the equivariant cohomology of the fixed point subset. A report should distinguish that evidence from a proxy, consequence, or common implementation. It should also state the qualification This does not extend, in verbatim, to the non-abelian action. so that a reader can reproduce the classification rather than infer it from topical resemblance.

Manages Complexity¶

Localization formula for equivariant cohomology compresses multiple mathematics, logic, and statistics details into a stable diagnostic relation. The source shows both the central mechanism—where H is Hamiltonian for the circle action, the sum is over points fixed by the circle action and \alpha_j(p) are eigenvalues on the tangent space at p (cf.—and the practical consequence—this does not extend, in verbatim, to the non-abelian action. This compression makes cases comparable while leaving parameters, conventions, exceptions, and evidential quality explicit. It is lossy by design: local history and implementation details may be omitted only when they do not alter the defining relation.

Abstract Reasoning¶

Type the carrier. Identify the mathematics, logic, and statistics entities to which the claim applies.
State the relation. Use the source-grounded identity: The localization theorem for equivariant cohomology in non-rational coefficients is discussed in Daniel Quillen's papers.
Check operation and conditions. An alternative name for the formula is Borel cohomology, after Armand Borel.
Demand recognition evidence. The localization theorem states that the equivariant cohomology can be recovered, up to torsion elements, from the equivariant cohomology of the fixed point subset.
Test variation. Change an implementation or setting while preserving but, there is still a version of the localization theorem for non-abelian actions.
Run the collapse test. Remove the defining operation; if the label still seems equally apt, only a topic or correlate was retained.
Reduce cautiously. When the specialist conditions cannot be carried, route the residual comparison to Representation.

Knowledge Transfer¶

Within the home domain. Knowledge about Localization formula for equivariant cohomology transfers literally when a new case preserves the same carrier type, relation, and recognition test. The formula allows one to compute the equivariant cohomology ring of the orbifold M (a particular kind of differentiable stack) from the equivariant cohomology of its fixed point components, up to multiplicities and Euler forms. An alternative name for the formula is Borel cohomology, after Armand Borel.

Beyond the home domain. No canonical parent is asserted for Localization formula for equivariant cohomology. An outside case receives the specialist name only when the same typed roles and rejection conditions can be filled literally; otherwise the comparison remains an analogy pending later graph densification.

Examples¶

Canonical¶

An alternative name for the formula is Borel cohomology, after Armand Borel. This case is canonical because it supplies a concrete carrier and lets the defining relation be checked rather than merely named.

Mapped back: carrier → the entities in the documented case; operation → The localization theorem for equivariant cohomology in non-rational coefficients is discussed in Daniel Quillen's papers; recognition evidence → The localization theorem states that the equivariant cohomology can be recovered, up to torsion elements, from the equivariant cohomology of the fixed point subset

Applied / In Practice¶

The localization theorem states that the equivariant cohomology can be recovered, up to torsion elements, from the equivariant cohomology of the fixed point subset. The applied case shows how the identity is used under a second setting or qualification while keeping the same operative relation.

Mapped back: changed setting → Non-abelian localization; invariant → The localization theorem for equivariant cohomology in non-rational coefficients is discussed in Daniel Quillen's papers; boundary → the case exits the class when this does not extend, in verbatim, to the non-abelian action

Structural Tensions¶

T1 — Stable identity versus admissible variation. This does not extend, in verbatim, to the non-abelian action. The tension matters because emphasizing only one side either dissolves the identity or overstates what the evidence and domain conventions warrant.

Diagnostic: Which changes preserve the defining relation, and which replace it?

T2 — Recognition versus proxy. In differential geometry, the localization formula states that for an equivariantly closed equivariant differential form \alpha on an orbifold M with a torus action and for a sufficient small \xi in the Lie algebra of the torus T, we have. The tension matters because emphasizing only one side either dissolves the identity or overstates what the evidence and domain conventions warrant.

Diagnostic: Does the cited evidence establish the identity or only a correlated sign?

T3 — Definition versus implementation. The formula allows one to compute the equivariant cohomology ring of the orbifold M (a particular kind of differentiable stack) from the equivariant cohomology of its fixed point components, up to multiplicities and Euler forms. The tension matters because emphasizing only one side either dissolves the identity or overstates what the evidence and domain conventions warrant.

Diagnostic: Is the observed implementation constitutive, optional, or merely common?

T4 — Scope versus overextension. An alternative name for the formula is Borel cohomology, after Armand Borel. The tension matters because emphasizing only one side either dissolves the identity or overstates what the evidence and domain conventions warrant.

Diagnostic: Can every claimed application fill the same typed roles without metaphor?

T5 — Transfer versus domain accent. The formula allows one to compute the equivariant cohomology ring of the orbifold M (a particular kind of differentiable stack) from the equivariant cohomology of its fixed point components, up to multiplicities and Euler forms. The tension matters because emphasizing only one side either dissolves the identity or overstates what the evidence and domain conventions warrant.

Diagnostic: Does the receiving case instantiate Localization formula for equivariant cohomology literally, co-instantiate Representation, or only resemble it?

T6 — Autonomy versus reduction. where H is Hamiltonian for the circle action, the sum is over points fixed by the circle action and \alpha_j(p) are eigenvalues on the tangent space at p (cf. The tension matters because emphasizing only one side either dissolves the identity or overstates what the evidence and domain conventions warrant.

Diagnostic: What does Localization formula for equivariant cohomology distinguish that the broader parent Representation leaves together?

Structural–Framed Character¶

Localization formula for equivariant cohomology is structural-leaning. Its structural side is the repeatable organization summarized by The localization theorem for equivariant cohomology in non-rational coefficients is discussed in Daniel Quillen's papers. Its framed side is the mathematics, logic, and statistics vocabulary that fixes the carrier, evidence, exceptions, and admissible transformations.

Evaluative weight: the identity can be stated descriptively even when applications carry practical stakes. Human-practice dependence: the source-grounded carrier determines whether the relation exists independently or is constituted by a practice. Institutional origin: disciplinary conventions stabilize the name and test. Vocabulary portability: An alternative name for the formula is Borel cohomology, after Armand Borel. Import versus recognition: literal transfer requires the same mechanism; shape alone is analogy.

Its portable skeleton is Representation. Its character: a recurring specialist identity whose thin organization can be abstracted, while its operational meaning remains domain-bound.

Structural Core vs. Domain Accent¶

What is skeletal. The localization theorem for equivariant cohomology in non-rational coefficients is discussed in Daniel Quillen's papers. The stable skeleton is the typed relation expressed in that definition and the entry's recognition and collapse tests. The source identifies these operative conditions: The formula allows one to compute the equivariant cohomology ring of the orbifold M (a particular kind of differentiable stack) from the equivariant cohomology of its fixed point components, up to multiplicities and Euler forms. where H is Hamiltonian for the circle action, the sum is over points fixed by the circle action and \alphaj(p) are eigenvalues on the tangent space at p (cf. It further constrains recognition and variation through: An alternative name for the formula is Borel cohomology, after Armand Borel. The localization theorem states that the equivariant cohomology can be recovered, up to torsion elements, from the equivariant cohomology of the fixed point subset.

What is domain-bound. mathematics, logic, and statistics supplies the operative entities, technical vocabulary, warrants, and exceptions that make Localization formula for equivariant cohomology literal. Its documented scope includes the condition that The formula allows one to compute the equivariant cohomology ring of the orbifold M (a particular kind of differentiable stack) from the equivariant cohomology of its fixed point components, up to multiplicities and Euler forms. Another bounded application condition is that An alternative name for the formula is Borel cohomology, after Armand Borel. These are not decorative examples; they determine which carrier and evidence can fill the abstraction's roles.

Why no parent is asserted. Removing those specialist details does not currently yield one live catalog node that is a necessary genus for every instance. The entry is therefore approved as unparented rather than attached by topical resemblance. Its collapse evidence remains specific—But, there is still a version of the localization theorem for non-abelian actions.—and future graph densification may discover a defensible relation only if it preserves that boundary.

Approved unparented node. No current live node supplies a defensible necessary genus or structural prerequisite for Localization formula for equivariant cohomology. The reviewed identity is: The localization theorem for equivariant cohomology in non-rational coefficients is discussed in Daniel Quillen's papers. The accelerated suggestion was declined because topical or lexical similarity does not establish hierarchy; the node is admitted without a parent pending later graph densification.
Related reasoning operations. Evidence, representation, comparison, classification, transformation, or evaluation may participate in particular cases, but participation does not make any one of them a necessary parent of every instance.

Neighborhood in Abstraction Space¶

Localization formula for equivariant cohomology sits in a moderately populated region (53^rd percentile for distinctiveness): it has near-neighbors but no dense thicket of look-alikes.

Family — Algebraic Structures, Groups & Operators (42 abstractions)

Nearest neighbors

Computed from structural-signature embeddings · 2026-10-08

Not to Be Confused With¶

Representation. The parent omits the specialist differentia. Tell: Can the case establish The localization theorem for equivariant cohomology in non-rational coefficients is discussed in Daniel Quillen's papers?
Verdier duality. A derived-sheaf duality that exchanges proper direct image with exceptional inverse image and extends Poincaré duality to singular spaces and maps. Tell: Which entry's carrier, operation, and failure condition are satisfied?
Equivariant differential form. A group-equivariant polynomial map from a Lie algebra to differential forms on a manifold, representing a cochain in the Cartan model of equivariant cohomology. Tell: Which entry's carrier, operation, and failure condition are satisfied?
Euler sequence. A canonical exact sequence of sheaves on projective space relating relative differentials to twists of the structure sheaf. Tell: Which entry's carrier, operation, and failure condition are satisfied?
A measurement, proxy, or consequence. Those may provide evidence without being the identity. Tell: Would Localization formula for equivariant cohomology remain present if the detector or downstream effect changed?
A metaphorical analogue. A similar shape outside mathematics, logic, and statistics lacks the specialist mechanism. Tell: Do the native roles transfer literally, or only the parent Representation?

References¶

Frozen Wikipedia discovery revision: https://en.wikipedia.org/wiki/Localization_formula_for_equivariant_cohomology (revision 1342189863).

The frozen Wikipedia revision is discovery provenance. The retained source set was reviewed for identity, formal or operational relation, and scope. The encyclopedia's structural synthesis is bounded to those claims; a thin authority surface is recorded as a nonblocking source-strengthening repair rather than concealed.