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


Author: Barry Jessup
Journal: Trans. Amer. Math. Soc. 317 (1990), 655-660
MSC: Primary 55P62; Secondary 55P50
DOI: https://doi.org/10.1090/S0002-9947-1990-0956033-8
MathSciNet review: 956033
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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.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1990-0956033-8
Keywords: Lusternik-Schnirelmann category, minimal models, rational homotopy
Article copyright: © Copyright 1990 American Mathematical Society

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