Two special cases of Ganea's conjecture

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.