Rational approximations to L-S category and a conjecture of Ganea

Abstract:

The rational version of Ganea’s conjecture for L-S category, namely that $\operatorname {cat} (S \times {\Sigma ^k}) = \operatorname {cat} (S) + 1$, if $S$ is a rational space and ${\Sigma ^k}$ denotes the $k$-sphere, is still open. Recently, a module type approximation to $\operatorname {cat} (S)$, was introduced by Halperin and Lemaire. We have previously shown that $M\operatorname {cat}$ satisfies Ganea’s conjecture. Here we show that for $(r - 1)$ connected $S$, if $M\operatorname {cat} (S)$ is at least $\dim S/2r$, then $M\operatorname {cat} (S) = \operatorname {cat} (S)$. This yields Ganea’s conjecture for these spaces. We also extend other properties of $M\operatorname {cat}$, previously unknown for cat, to these spaces.