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If C=C_\phi denotes the mapping cone of an essential phantom map \phi fromthe suspension of the Eilenberg-Mac Lane complex K=K(Z,5) to the 4-sphere S=S^4we derive the following properties: (1) The LS category of the product of Cwith any n-sphere S^n is equal to 3; (2) The LS category of the product of Cwith itself is equal to 3, hence is strictly less than twice the LS category ofC. These properties came to light in the course of an unsuccessful attempt tofind, for each positive integer m, an example of a pair of 1-connectedCW-complexes of finite type in the same Mislin (localization) genus with LScategories m and 2m. If \phi is such that its p-localizations are inessentialfor all primes p, then by the main result of [J. Roitberg, TheLusternik-Schnirelmann category of certain infinite CW-complexes, Topology 39(2000), 95-101], the pair C_*, C where C_*= S wedge \Sigma ^2 K, provides suchan example in the case m=1.

The product formula for Lusternik–Schnirelmann category