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A topological space has Lusternik–Schnirelmann category if there is an open cover of by at most sets for which the inclusion maps are nullhomotopic. The Lusternik–Schnirelmann category or LS-category of is the least such that has LS-category .
Parts of the literature use an alternative definition, according to which the LS-category is one less than the definition used here.
The LS-category is a homotopy invariant: if is a homotopy equivalence and an open cover of such that the inclusions are nullhomotopic, then is an open cover for which the each inclusion is nullhomotopic, since its composition with factors through the nullhomotopic map .
A space has LS-category iff it is empty, and LS-category iff it is contractible. For discrete spaces the LS-category is equal to the cardinality. Any suspension has LS-category ; in particular the sphere has LS-category if .
For path-connected spaces and , the LS-category of is exactly the sectional category? of the path fibration .
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]
Last revised on July 20, 2025 at 08:48:54. See the history of this page for a list of all contributions to it.