Leray–Schauder degree
E518467
The Leray–Schauder degree is a topological invariant that generalizes the Brouwer degree to compact perturbations of the identity in infinite-dimensional Banach spaces, providing a powerful tool for proving existence of solutions to nonlinear equations.
All labels observed (1)
| Label | Occurrences |
|---|---|
| Leray–Schauder degree canonical | 2 |
How this entity was disambiguated
This entity first appeared as the object of triple T5425419 — resolving that mention is where its identity was fixed. The disambiguator weighed these candidate entities and picked the highlighted one (or “None”, minting a new entity). This is how homonymy is resolved: the same surface form can point to different entities.
NED1
Entity disambiguation (via context triple)
gpt-5-mini-2025-08-07
Target entity: Leray–Schauder degree Context triple: [Schauder fixed-point theorem, relatedTo, Leray–Schauder degree]
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A.
Schauder fixed-point theorem
The Schauder fixed-point theorem is a fundamental result in functional analysis that guarantees the existence of fixed points for continuous compact mappings on convex closed subsets of Banach spaces, generalizing the Brouwer fixed-point theorem to infinite-dimensional settings.
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B.
Lefschetz fixed-point theorem
The Lefschetz fixed-point theorem is a fundamental result in algebraic topology that relates the number of fixed points of a continuous map on a topological space to traces of the induced maps on its homology groups.
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C.
Brouwer fixed-point theorem
The Brouwer fixed-point theorem is a fundamental result in topology stating that any continuous function from a compact convex set (such as a closed disk) to itself has at least one fixed point.
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D.
Atiyah–Bott fixed-point theorem
The Atiyah–Bott fixed-point theorem is a fundamental result in equivariant cohomology that expresses global invariants, such as indices of elliptic operators, in terms of local data at the fixed points of a group action.
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E.
Poincaré–Hopf theorem
The Poincaré–Hopf theorem is a fundamental result in differential topology that relates the sum of the indices of a vector field’s isolated zeros on a compact manifold to the manifold’s Euler characteristic.
- F. None of above. chosen
- G. Unsure - the case is ambiguous/there is not enough information to decide.
NED2
Entity disambiguation (via description)
gpt-5-mini-2025-08-07
Target entity: Leray–Schauder degree Target entity description: The Leray–Schauder degree is a topological invariant that generalizes the Brouwer degree to compact perturbations of the identity in infinite-dimensional Banach spaces, providing a powerful tool for proving existence of solutions to nonlinear equations.
-
A.
Schauder fixed-point theorem
The Schauder fixed-point theorem is a fundamental result in functional analysis that guarantees the existence of fixed points for continuous compact mappings on convex closed subsets of Banach spaces, generalizing the Brouwer fixed-point theorem to infinite-dimensional settings.
-
B.
Lefschetz fixed-point theorem
The Lefschetz fixed-point theorem is a fundamental result in algebraic topology that relates the number of fixed points of a continuous map on a topological space to traces of the induced maps on its homology groups.
-
C.
Brouwer fixed-point theorem
The Brouwer fixed-point theorem is a fundamental result in topology stating that any continuous function from a compact convex set (such as a closed disk) to itself has at least one fixed point.
-
D.
Atiyah–Bott fixed-point theorem
The Atiyah–Bott fixed-point theorem is a fundamental result in equivariant cohomology that expresses global invariants, such as indices of elliptic operators, in terms of local data at the fixed points of a group action.
-
E.
Poincaré–Hopf theorem
The Poincaré–Hopf theorem is a fundamental result in differential topology that relates the sum of the indices of a vector field’s isolated zeros on a compact manifold to the manifold’s Euler characteristic.
- F. None of above. chosen
Statements (44)
| Predicate | Object |
|---|---|
| instanceOf |
nonlinear analysis tool
ⓘ
topological degree ⓘ topological invariant ⓘ |
| appliesTo |
compact perturbations of the identity
ⓘ
infinite-dimensional Banach spaces ⓘ maps in Banach spaces ⓘ |
| assumes | Fredholm-type structure in many applications ⓘ |
| codomain | integers ⓘ |
| comparedTo | Brouwer degree ⓘ |
| definedFor | compact maps on open bounded subsets of Banach spaces ⓘ |
| domain | Banach space ⓘ |
| field |
functional analysis
ⓘ
nonlinear functional analysis ⓘ partial differential equations ⓘ topology ⓘ |
| generalizes | Brouwer degree ⓘ |
| hasVariant |
degree for Fredholm maps of index zero
ⓘ
degree for condensing maps ⓘ |
| historicalPeriod | 20th century mathematics ⓘ |
| namedAfter |
Jean Leray
ⓘ
Julius Schauder ⓘ |
| property |
coincides with Brouwer degree in finite dimensions
ⓘ
homotopy invariant ⓘ integer-valued ⓘ stable under compact perturbations ⓘ |
| relatedTo |
Leray–Schauder fixed point theorem
ⓘ
linked to:
Schauder fixed-point theorem
Schauder fixed point theorem ⓘ topological degree theory ⓘ |
| requires |
a priori bounds on solutions
ⓘ
compactness of the nonlinear part ⓘ |
| satisfies |
additivity property
ⓘ
excision property ⓘ homotopy invariance property ⓘ normalization property ⓘ |
| toolFor |
integral equations
ⓘ
nonlinear boundary value problems ⓘ nonlinear operator equations ⓘ |
| usedFor |
boundary value problems
ⓘ
elliptic partial differential equations ⓘ existence of solutions of nonlinear equations ⓘ fixed point problems ⓘ |
| usedIn |
continuation methods
ⓘ
degree-theoretic proofs of existence theorems ⓘ global bifurcation theory ⓘ |
How these facts were elicited
The pipeline generated the facts above by prompting gpt-5.1 with this entity's name + description and the instruction below.
Instruction
You are a knowledge base construction expert. Given a subject entity and a description of it, return factual statements that you know for the subject as a JSON list of dictionaries(triples), where keys must be "subject", "predicate" and "object". The number of facts may be very high, between 25 to 50 or more, for very popular subjects. For less popular subjects, the number of facts can be very low, like 5 or 10. # Requirements - If you don't know the subject at all, return an empty list. - If the subject is not a named entity, return an empty list. - Include at least one triple where predicate is "instanceOf". - Do not get too wordy. - Separate several objects into multiple triples with one object.
Input
Subject: Leray–Schauder degree Description of subject: The Leray–Schauder degree is a topological invariant that generalizes the Brouwer degree to compact perturbations of the identity in infinite-dimensional Banach spaces, providing a powerful tool for proving existence of solutions to nonlinear equations.
Referenced by (2)
Full triples — surface form annotated when it differs from this entity's canonical label.