The product formula for Lusternik–Schnirelmann category

If $C = C_{ϕ}$ , 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 $S = S^{4}$ , 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 $2 m .$ 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 $C_{*} = S \lor Σ^{2} K, C$ provides such an example in the case $m = 1$ .

Keywords

phantom map, Mislin (localization) genus, Lusternik–Schnirelmann category, Hopf invariant, cuplength

Mathematical Subject Classification 2000

Primary: 55M30

References

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Publication

Received: 26 October 2000

Revised: 7 May 2001

Accepted: 17 August 2001

Published: 10 September 2001

Authors

Joseph Roitberg
Department of Mathematics and Statistics Hunter Colleg CUNY 695 Park Avenue New York NY 10021 USA

Volume 1, issue 1 (2001)

Joseph Roitberg