The product formula for Lusternik–Schnirelmann category

Roitberg, Joseph¹

¹Department of Mathematics and Statistics, Hunter Colleg, CUNY, 695 Park Avenue, New York NY 10021, USA

Algebraic and Geometric Topology, Tome 1 (2001) no. 1, pp. 491-502

Résumé

If C = Cϕ, denotes the mapping cone of an essential phantom map ϕ from the suspension of the Eilenberg–Mac Lane complex K = K(ℤ,5), to the 4–sphere S = S4, we derive the following properties: (1) The LS category of the product of C with any n–sphere Sn is equal to 3; (2) The LS category of the product of C with itself is equal to 3, hence is strictly less than twice the LS category of C. These properties came to light in the course of an unsuccessful attempt to find, for each positive integer m, an example of a pair of 1–connected CW–complexes of finite type in the same Mislin (localization) genus with LS categories m and 2m. If ϕ is such that its p–localizations are inessential for all primes p, then by the main result of [J. Roitberg, The Lusternik–Schnirelmann category of certain infinite CW–complexes, Topology 39 (2000), 95–101], the pair C∗ = S ∨ Σ2K,C provides such an example in the case m = 1.

Détail

DOI : 10.2140/agt.2001.1.491

Keywords: phantom map, Mislin (localization) genus, Lusternik–Schnirelmann category, Hopf invariant, cuplength

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