Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Rational LS category and its applications

Authors: Yves Félix and Stephen Halperin
Journal: Trans. Amer. Math. Soc. 273 (1982), 1-37
MSC: Primary 55P62; Secondary 55M30, 55P50
MathSciNet review: 664027
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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$.

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Keywords: Lusternik-Schnirelmann category, minimal model, rational homotopy, rank of a space
Article copyright: © Copyright 1982 American Mathematical Society