Algebraic & Geometric Topology

The product formula for Lusternik–Schnirelmann category

Joseph Roitberg

Abstract

If $C=Cϕ$, denotes the mapping cone of an essential phantom map $ϕ$ from the suspension of the Eilenberg–Mac Lane complex $K=K(ℤ,5)$, 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 $m=1$.

Article information

Source
Algebr. Geom. Topol., Volume 1, Number 1 (2001), 491-502.

Dates
Revised: 7 May 2001
Accepted: 17 August 2001
First available in Project Euclid: 21 December 2017

https://projecteuclid.org/euclid.agt/1513882605

Digital Object Identifier
doi:10.2140/agt.2001.1.491

Mathematical Reviews number (MathSciNet)
MR1852769

Zentralblatt MATH identifier
0976.55002

Citation

Roitberg, Joseph. The product formula for Lusternik–Schnirelmann category. Algebr. Geom. Topol. 1 (2001), no. 1, 491--502. doi:10.2140/agt.2001.1.491. https://projecteuclid.org/euclid.agt/1513882605

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