nLab Lusternik-Schnirelmann category

Context

Homotopy theory

homotopy theory, (∞,1)-category theory, homotopy type theory

flavors: stable, equivariant, rational, p-adic, proper, geometric, cohesive, directed

models: topological, simplicial, localic, …

see also algebraic topology

Introductions

Definitions

Paths and cylinders

Homotopy groups

Basic facts

Theorems

Contents

Definition

Definition

A topological space XX has Lusternik–Schnirelmann category n \le n if there is an open cover of XX by at most nn sets for which the inclusion maps UX U \hookrightarrow X are nullhomotopic. The Lusternik–Schnirelmann category or LS-category of XX is the least nn such that XX has LS-category n\le n.

Remark

Parts of the literature use an alternative definition, according to which the LS-category is one less than the definition used here.

Properties

  • The LS-category is a homotopy invariant: if f:XY f\colon X \to Y is a homotopy equivalence and {U α} 0α<n \{ U_\alpha \}_{0 \le \alpha \lt n} an open cover of YY such that the inclusions i α:UYi_\alpha \colon U\hookrightarrow Y are nullhomotopic, then {f 1(U α)} 0α<n\{f^{-1}(U_\alpha)\}_{0\le\alpha \lt n} is an open cover for which the each inclusion f 1(U α)Xf^{-1}(U_\alpha) \hookrightarrow X is nullhomotopic, since its composition with ff factors through the nullhomotopic map i αi_\alpha.

  • A space has LS-category 00 iff it is empty, and LS-category 11 iff it is contractible. For discrete spaces the LS-category is equal to the cardinality. Any suspension has LS-category 22; in particular the sphere S nS^n has LS-category 22 if n0n \ge 0.

  • For path-connected spaces XX and x 0Xx_0\in X, the LS-category of XX is exactly the sectional category? of the path fibration ev 1:{γX I:γ(0)=x 0}Xev_1 \colon \{\gamma \in X^I : \gamma(0)=x_0\}\to X.

References

Last revised on July 20, 2025 at 08:48:54. See the history of this page for a list of all contributions to it.