Rational LS category and its applications

Abstract

Let S S be a 1 1 -connected CW-complex of finite type and put ca t 0 ( S ) = {\text {ca}}{{\text {t}}_0}(S) = Lusternik-Schnirelmann category of the localization S Q {S_{\mathbf {Q}}} . This invariant is characterized in terms of the minimal model of S S . It is shown that if ϕ : S → T \phi :S \to T is injective on π ∗ ⊗ Q {\pi _ \ast } \otimes {\mathbf {Q}} then ca t 0 ( S ) ⩽ ca t 0 ( T ) {\text {ca}}{{\text {t}}_0}(S) \leqslant {\text {ca}}{{\text {t}}_0}(T) , and this result is strengthened when ϕ \phi is the fibre inclusion of a fibration. It is also shown that if H ∗ ( S ; Q ) > ∞ {H^ \ast }(S;{\mathbf {Q}}) > \infty then either π ∗ ( S ) ⊗ Q > ∞ {\pi _ \ast }(S) \otimes {\mathbf {Q}} > \infty or the groups π k ( S ) ⊗ Q {\pi _k}(S) \otimes {\mathbf {Q}} grow exponentially with k k .