Two special cases of Ganea’s conjecture

Abstract:

Ganea conjectured that for any finite CW complex $X$ and any $k>0$, $\operatorname {cat}(X\times S^k) =\operatorname {cat}(X) + 1$. In this paper we prove two special cases of this conjecture. The main result is the following. Let $X$ be a $(p-1)$-connected $n$-dimensional CW complex (not necessarily finite). We show that if $\operatorname {cat}(X) = \left \lfloor {n \over p} \right \rfloor + 1$ and $n\not \equiv -1 \operatorname {mod} p$ (which implies $p>1$), then $\operatorname {cat}(X\times S^k) =\operatorname {cat}(X) +1$. This is proved by showing that $\operatorname {wcat}(X\times S^k) =\operatorname {wcat}(X) + 1$ in a much larger range, and then showing that under the conditions imposed, $\operatorname {cat}(X)=\operatorname {wcat}(X)$. The second special case is an extension of Singhof’s earlier result for manifolds.