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The product formula for Lusternik–Schnirelmann category

Joseph Roitberg

Algebraic & Geometric Topology 1 (2001) 491–502

arXiv: math.AT/0109105


If C = Cϕ, denotes the mapping cone of an essential phantom map ϕ from the suspension of the Eilenberg–Mac Lane complex K = K(,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.

phantom map, Mislin (localization) genus, Lusternik–Schnirelmann category, Hopf invariant, cuplength
Mathematical Subject Classification 2000
Primary: 55M30
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Received: 26 October 2000
Revised: 7 May 2001
Accepted: 17 August 2001
Published: 10 September 2001
Joseph Roitberg
Department of Mathematics and Statistics
Hunter Colleg
695 Park Avenue
New York NY 10021