Lusternik–Schnirelmann category of products with half-smashes

Stanley, Don; Strom, Jeff

doi:10.2140/agt.2020.20.439

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Lusternik–Schnirelmann category of products with half-smashes

Don Stanley and Jeff Strom

Algebraic & Geometric Topology 20 (2020) 439–450

DOI: 10.2140/agt.2020.20.439

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 \times A) = cat (X \times (A ⋊ B)) for all spaces B .

Taking $X = S^{n}$ leads to a closure property for the collections of spaces which do or do not satisfy the Ganea condition $cat (S^{n} \times A) = 1 + cat (A)$ .

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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

Volume 20, issue 1 (2020)