Algebraic & Geometric Topology 1 (2001) 491–502

DOI: 10.2140/agt.2001.1.491

arXiv: math.AT/0109105

Abstract

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=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 the pair C_* = S∨Σ²K, 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

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