We define and study the categorical sequence of a space, which is a new formalism that streamlines the computation of the Lusternik-Schnirelmann category of a space X by induction on its CW skeleta. The k th term in the categorical sequence of a CW complex X , X .k/, is the least integer n for which cat X .X n / k . We show that X is a well-defined homotopy invariant of X . We prove that X .kCl/ X .k/C X .l/, which is one of three keys to the power of categorical sequences. In addition to this formula, we provide formulas relating the categorical sequences of spaces and some of their algebraic invariants, including their cohomology algebras and their rational models; we also find relations between the categorical sequences of the spaces in a fibration sequence and give a preliminary result on the categorical sequence of a product of two spaces in the rational case. We completely characterize the sequences which can arise as categorical sequences of formal rational spaces. The most important of the many examples that we offer is a simple proof of a theorem of Ghienne: if X is a member of the Mislin genus of the Lie group Sp.3/, then cat.X / D cat.Sp.3//. 55M30; 55P62 IntroductionThe Lusternik-Schnirelmann category of a topological space X is the least integer k for which X has an open cover X D X 0 [ X 1 [ [ X k with the property that each inclusion map X j ,! X is homotopic to a constant map; it is denoted cat.X /. This homotopy invariant of topological spaces was first introduced by Lusternik and Schnirelmann in 1934 as a tool to use in studying functions on (compact) manifolds: a smooth function f W M ! ޒ must have at least cat.M / C 1 critical points.If X is a CW complex, then X n D X n 1 [˛.n cells/, and therefore cat.X n / Ä cat.X n 1 / C 1. Berstein and Hilton asked [3] what conditions must be placed on the attaching map˛in order to guarantee that equality holds in this upper bound; the answer is that equality holds when a certain set of generalized Hopf invariants does This is done via the categorical sequence of a space X , which is a function X W ގ ! ގ [ f1g defined byX n / kg; where cat X .X n / is the category of X n relative to X (see Definition 4) 1 . It is shown in Propositions 2.1 and 2.2 that X is a well-defined homotopy invariant of X ; ie, when n is larger than the connectivity of X , cat X .X n / depends only on n and the homotopy type of X , and not on any choices made in constructing a CW decomposition of X . If X is finite-dimensional, then X determines cat.X /; examples due to Roitberg [28] show that this is not true for infinite-dimensional spaces. In any case, the categorical sequence of X holds a wealth of useful information.Though we are not directly concerned with the applications of Lusternik-Schnirelmann category to critical point theory in this paper, categorical sequences could play a useful role there. For example, in the study of the n-body problem, one is often interested in infinite-dimensional Sobolev spaces W ; in order to apply the Lusternik-Schnirelmann method in thi...
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