If , denotes the mapping cone of an essential phantom map from the suspension of the Eilenberg–Mac Lane complex , to the –sphere , we derive the following properties: (1) The LS category of the product of with any –sphere is equal to ; (2) The LS category of the product of with itself is equal to , hence is strictly less than twice the LS category of . These properties came to light in the course of an unsuccessful attempt to find, for each positive integer , an example of a pair of –connected CW–complexes of finite type in the same Mislin (localization) genus with LS categories and If is such that its –localizations are inessential for all primes , then by the main result of [J. Roitberg, The Lusternik–Schnirelmann category of certain infinite CW–complexes, Topology 39 (2000), 95–101], the pair provides such an example in the case .
"The product formula for Lusternik–Schnirelmann category." Algebr. Geom. Topol. 1 (1) 491 - 502, 2001. https://doi.org/10.2140/agt.2001.1.491