Volume 4, issue 2 (2004)

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Implications of the Ganea condition

Norio Iwase, Donald Stanley and Jeffrey Strom

Algebraic & Geometric Topology 4 (2004) 829–839
 arXiv: math.AT/0410374
Abstract

Suppose the spaces $X$ and $X×A$ have the same Lusternik–Schnirelmann category: $cat\left(X×A\right)=cat\left(X\right)$. Then there is a strict inequality $cat\left(X×\left(A⋊B\right)\right) for every space $B$, provided the connectivity of $A$ is large enough (depending only on $X$). This is applied to give a partial verification of a conjecture of Iwase on the category of products of spaces with spheres.

Keywords
Lusternik–Schnirelmann category, Ganea conjecture, product formula, cone length
Primary: 55M30