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Implications of the Ganea Condition. (English) Zbl 1067.55002

In 1971 T. Ganea asked whether \(cat(X \times S^r) =cat(X)+1\). This is not always true, but the question for which spaces \(Y\) \(cat(X \times Y) =cat(X)+cat(Y)\) is not easy to answer. The following results are proved.
Theorem 1. Let \(X\) be a \((c-1)\)-connected space and \(r > \dim(X)-c \cdot cat(X)+2\). If \(cat(X \times S^r)=cat(X)\), then \(cat(X \times S^t) = cat(X)\) for all \(t \geq r\).
Theorem 2. Let \(X\) be \((c-1)\)-connected and let \(A\) be \((r-1)\)-connected with \(r>dim(X)-c \cdot cat(X)+2\). If \(cat(X \times A)=cat(X)\), then \(cat(X \times ((A \times B)/B)) < cat(X) + cat((A \times B)/B))\) for every space \(B\).

MSC:

55M30 Lyusternik-Shnirel’man category of a space, topological complexity à la Farber, topological robotics (topological aspects)
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References:

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