#### Algebraic and Geometric Topology 1 (2001),
paper no. 25, pages 491-502.

## The product formula for Lusternik-Schnirelmann category

### Joseph Roitberg

**Abstract**.
If C=C_\phi denotes the mapping cone of an essential phantom map \phi
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 2m. If \phi 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_*, C where C_*= S wedge \Sigma
^2 K, provides such an example in the case m=1.
**Keywords**.
Phantom map, Mislin (localization) genus, Lusternik-Schnirelmann category,
Hopf invariant, Cuplength

**AMS subject classification**.
Primary: 55M30.

**DOI:** 10.2140/agt.2001.1.491

**E-print:** `arXiv:math.AT/0109105`

Submitted: 26 October 2000.
(Revised: 7 May 2001.)
Accepted: 17 August 2001.
Published: 10 September 2001.

Notes on file formats
Joseph Roitberg

Department of Mathematics and Statistics

Hunter College, CUNY

695 Park Avenue, New York, NY 10021, USA

Email: roitberg@math.hunter.cuny.edu

AGT home page

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