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Lusternik–Schnirelmann category of products with half-smashes

Don Stanley and Jeff Strom

Algebraic & Geometric Topology 20 (2020) 439–450
Abstract

We show that for a fixed space X and any sufficiently highly connected space A ( conn(A) > dim(X) is more than enough), the Lusternik–Schnirelmann category of products with X is remarkably stable with respect to changes in the second variable:

cat(X × A) = cat(X × (A B)) for all spaces B.

Taking X = Sn leads to a closure property for the collections of spaces which do or do not satisfy the Ganea condition cat(Sn × A) = 1 + cat(A).

Keywords
Lusternik–Schnirelmann category, half-smash, Ganea conjecture, Ganea condition, cartesian product, homotopy theory
Mathematical Subject Classification 2010
Primary: 55M30
References
Publication
Received: 4 December 2018
Revised: 8 May 2019
Accepted: 20 June 2019
Published: 23 February 2020
Authors
Don Stanley
Department of Mathematics and Statistics
University of Regina
Regina, SK
Canada
Jeff Strom
Department of Mathematics
Western Michigan University
Kalamazoo, MI
United States