Lusternik–Schnirelmann category

E687582

homotopy invariant invariant of topological spaces numerical invariant topological invariant

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.

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Observed surface forms (2)

Surface form	Occurrences
Lusternik–Schnirelmann theorem	1
Lusternik–Schnirelmann theory	1

Statements (47)

Predicate	Object
instanceOf	homotopy invariant ⓘ invariant of topological spaces ⓘ numerical invariant ⓘ topological invariant ⓘ
alsoKnownAs	LS category ⓘ Lusternik–Schnirelman category NERFINISHED ⓘ
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 NERFINISHED ⓘ
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 NERFINISHED ⓘ path-connected topological spaces ⓘ smooth manifolds ⓘ
measures	minimal number of contractible open sets in X needed to cover X ⓘ
namedAfter	Lazar Aronovich Lusternik NERFINISHED ⓘ Lev Genrikhovich Schnirelmann NERFINISHED ⓘ
relatedTo	Ganea fibrations NERFINISHED ⓘ Whitehead category NERFINISHED ⓘ category weight in cohomology ⓘ cup-length in cohomology ⓘ fibration category of a map ⓘ topological complexity of Farber ⓘ
symbol	cat(X) ⓘ
usedIn	Morse theory NERFINISHED ⓘ critical point theory of smooth functions ⓘ topological complexity theory ⓘ variational problems ⓘ
usedToBound	number of critical points of smooth maps on manifolds ⓘ

Referenced by (5)

Full triples — surface form annotated when it differs from this entity's canonical label.

Lazar Lyusternik → knownFor → Lusternik–Schnirelmann category ⓘ

this entity surface form: Lusternik–Schnirelmann theory

Lazar Lyusternik → coDeveloped → Lusternik–Schnirelmann category ⓘ

Lazar Lyusternik → notableConcept → Lusternik–Schnirelmann category ⓘ

this entity surface form: Lusternik–Schnirelmann theorem