Rational LS category and its applications

Abstract:

Let $S$ be a $1$-connected CW-complex of finite type and put ${\text {ca}}{{\text {t}}_0}(S) =$ Lusternik-Schnirelmann category of the localization ${S_{\mathbf {Q}}}$. This invariant is characterized in terms of the minimal model of $S$. It is shown that if $\phi :S \to T$ is injective on ${\pi _ \ast } \otimes {\mathbf {Q}}$ then ${\text {ca}}{{\text {t}}_0}(S) \leqslant {\text {ca}}{{\text {t}}_0}(T)$, and this result is strengthened when $\phi$ is the fibre inclusion of a fibration. It is also shown that if ${H^ \ast }(S;{\mathbf {Q}}) < \infty$ then either ${\pi _ \ast }(S) \otimes {\mathbf {Q}} < \infty$ or the groups ${\pi _k}(S) \otimes {\mathbf {Q}}$ grow exponentially with $k$.