nLab Lusternik-Schnirelmann category

Definition

Definition

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

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 $f\colon X \to Y$ is a homotopy equivalence and $\{ U_\alpha \}_{0 \le \alpha \lt n}$ an open cover of $Y$ such that the inclusions $i_\alpha \colon U\hookrightarrow Y$ are nullhomotopic, then $\{f^{-1}(U_\alpha)\}_{0\le\alpha \lt n}$ is an open cover for which the each inclusion $f^{-1}(U_\alpha) \hookrightarrow X$ is nullhomotopic, since its composition with $f$ factors through the nullhomotopic map $i_\alpha$.

• A space has LS-category $0$ iff it is empty, and LS-category $1$ iff it is contractible. For discrete spaces the LS-category is equal to the cardinality. Any suspension has LS-category $2$; in particular the sphere $S^n$ has LS-category $2$ if $n \ge 0$.

• For path-connected spaces $X$ and $x_0\in X$, the LS-category of $X$ is exactly the sectional category of the path fibration $ev_1 \colon \{\gamma \in X^I : \gamma(0)=x_0\}\to X$.

References

• Wikipedia, Lusternik-Schnirelman category

• I.M. James, On category, in the sense of Lusternik–Schnirelmann, Topology 17 4 (1978) 331–348 [doi:10.1016/0040-9383(78)90002-2]

• Ralph H. Fox, On the Lusternik–Schnirelmann category, Annals of Mathematics 42 2 (1941) 333–370 [doi:10.2307/1968905]