The product formula for Lusternik–Schnirelmann category

Abstract

If $C=C_{\varphi }$, denotes the mapping cone of an essential phantom map $\varphi$ from the suspension of the Eilenberg–Mac Lane complex $K=K\left(\mathbb{Z},5\right)$, to the $4$–sphere $S=S^4$, we derive the following properties: (1) The LS category of the product of $C$ with any $n$–sphere $S^n$ 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 $\varphi$ 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 $C_{\ast }=S\vee {\Sigma }^{2}K,C$ provides such an example in the case $m=1$.