Lusternik–Schnirelmann category
E687582
The Lusternik–Schnirelmann category is a numerical homotopy invariant of a topological space that measures the minimal number of contractible open sets needed to cover it, playing a key role in critical point theory and algebraic topology.
All labels observed (5)
| Label | Occurrences |
|---|---|
| Lusternik–Schnirelmann category canonical | 3 |
| Lusternik–Schnirelman category | 1 |
| Lusternik–Schnirelmann theorem | 1 |
| Lusternik–Schnirelmann theory | 1 |
| tangential Lusternik–Schnirelmann category | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T7770013 — 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: Lusternik–Schnirelmann category Context triple: [Lazar Lyusternik, knownFor, Lusternik–Schnirelmann category]
-
A.
Leray–Schauder degree
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.
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B.
Alexandrov–Čech cohomology
Alexandrov–Čech cohomology is a topological cohomology theory that computes invariants of spaces using inverse limits over open covers, closely related to and often coinciding with sheaf cohomology.
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C.
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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D.
Morse Theory
Morse Theory is a branch of differential topology that studies the relationship between the topology of manifolds and the critical points of smooth real-valued functions defined on them.
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E.
Moscow school of topology
The Moscow school of topology was a prominent mathematical tradition centered in Moscow that made foundational contributions to general and algebraic topology in the 20th century.
- 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: Lusternik–Schnirelmann category Target entity description: The Lusternik–Schnirelmann category is a numerical homotopy invariant of a topological space that measures the minimal number of contractible open sets needed to cover it, playing a key role in critical point theory and algebraic topology.
-
A.
Leray–Schauder degree
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.
-
B.
Alexandrov–Čech cohomology
Alexandrov–Čech cohomology is a topological cohomology theory that computes invariants of spaces using inverse limits over open covers, closely related to and often coinciding with sheaf cohomology.
-
C.
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.
-
D.
Morse Theory
Morse Theory is a branch of differential topology that studies the relationship between the topology of manifolds and the critical points of smooth real-valued functions defined on them.
-
E.
Moscow school of topology
The Moscow school of topology was a prominent mathematical tradition centered in Moscow that made foundational contributions to general and algebraic topology in the 20th century.
- F. None of above. chosen
Statements (47)
| Predicate | Object |
|---|---|
| instanceOf |
homotopy invariant
ⓘ
invariant of topological spaces ⓘ numerical invariant ⓘ topological invariant ⓘ |
| alsoKnownAs |
LS category
ⓘ
Lusternik–Schnirelman category ⓘ
linked to:
Lusternik–Schnirelmann category
|
| codomain | nonnegative integers ⓘ |
| definesFunctionOn | topological spaces ⓘ |
| field |
algebraic topology
ⓘ
critical point theory ⓘ homotopy theory ⓘ |
| generalizationOf | covering dimension in some contexts ⓘ |
| hasApplication |
existence of multiple periodic orbits in dynamical systems
ⓘ
lower bounds on number of critical points of smooth functions ⓘ multiplicity results in nonlinear analysis ⓘ |
| hasGeneralization |
equivariant Lusternik–Schnirelmann category
ⓘ
relative Lusternik–Schnirelmann category ⓘ tangential Lusternik–Schnirelmann category ⓘ
linked to:
Lusternik–Schnirelmann category
|
| hasInequality | cat(X) ≥ cup-length(X) + 1 ⓘ |
| hasProperty |
can be infinite for some spaces
ⓘ
cat(X) = 0 if and only if X is contractible ⓘ cat(X) = 1 for spheres S^n with n ≥ 1 ⓘ cat(X) ≤ dimension of X for reasonable spaces ⓘ finite for compact CW complexes ⓘ homotopy invariant of spaces ⓘ invariant under homotopy equivalence ⓘ monotone under maps that admit homotopy sections in some formulations ⓘ subadditive under products up to bounds ⓘ |
| introducedIn | 1930s ⓘ |
| isDefinedFor |
CW complexes
ⓘ
path-connected topological spaces ⓘ smooth manifolds ⓘ |
| measures | minimal number of contractible open sets in X needed to cover X ⓘ |
| namedAfter |
Lazar Aronovich Lusternik
ⓘ
linked to:
Lazar Lyusternik
Lev Genrikhovich Schnirelmann ⓘ |
| relatedTo |
Ganea fibrations
ⓘ
Whitehead category ⓘ category weight in cohomology ⓘ cup-length in cohomology ⓘ fibration category of a map ⓘ topological complexity of Farber ⓘ |
| symbol | cat(X) ⓘ |
| usedIn |
Morse theory
ⓘ
linked to:
Morse Theory
critical point theory of smooth functions ⓘ topological complexity theory ⓘ variational problems ⓘ |
| usedToBound | number of critical points of smooth maps on manifolds ⓘ |
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: Lusternik–Schnirelmann category Description of subject: The Lusternik–Schnirelmann category is a numerical homotopy invariant of a topological space that measures the minimal number of contractible open sets needed to cover it, playing a key role in critical point theory and algebraic topology.
Referenced by (7)
Full triples — surface form annotated when it differs from this entity's canonical label.
linked to: Lusternik–Schnirelmann category
linked to: Lusternik–Schnirelmann category
linked to: Lusternik–Schnirelmann category
linked to: Lusternik–Schnirelmann category