#### Volume 1, issue 1 (2001)

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The product formula for Lusternik–Schnirelmann category

### Joseph Roitberg

Algebraic & Geometric Topology 1 (2001) 491–502
 arXiv: math.AT/0109105
##### Abstract

If $C={C}_{\varphi }$, denotes the mapping cone of an essential phantom map $\varphi$ from the suspension of the Eilenberg–Mac Lane complex $K=K\left(ℤ,5\right)$, 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 ${C}_{\ast }=S\vee {\Sigma }^{2}K,\phantom{\rule{0.3em}{0ex}}C$ provides such an example in the case $m=1$.

##### Keywords
phantom map, Mislin (localization) genus, Lusternik–Schnirelmann category, Hopf invariant, cuplength
Primary: 55M30