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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Probing L-S category with maps


Author: Barry Jessup
Journal: Trans. Amer. Math. Soc. 339 (1993), 351-360
MSC: Primary 55M30; Secondary 55P60, 55P62
MathSciNet review: 1112375
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Abstract: For any map $ X\xrightarrow{f}Y$, we introduce two new homotopy invariants, $ {\text{dcat}}\;f$ and $ {\text{rcat}}\;f$. The classical category $ {\text{cat}}\;f$ is a lower bound for both, while $ {\text{dcat}}\;f \leq {\text{cat}}\;X$ and $ {\text{rcat}}\;f \leq {\text{cat}}\;Y$. When $ Y$ is an Eilenberg-Mac Lane space, $ f$ represents a cohomology class and $ {\text{dcat}}\;f$ often gives a good estimate for $ {\text{cat}}\;X$. We prove that if $ \Omega \in {H^n}(M;\mathbb{Z})$ is the fundamental class of a compact, simply connected $ n$-manifold, then $ {\text{dcat}}\;\Omega = {\text{cat}}\;M$. Similarly, when $ X$ is sphere, then $ f$ is a homotopy class and while $ {\text{cat}}\;f = 1$, $ {\text{rcat}}\;f$ can be a good approximation to $ {\text{cat}}\;Y$. We show that if $ \alpha \in {\pi _2}(\mathbb{C}{P^n})$ is nonzero, then $ {\text{rcat}}\;\alpha = n$. Rational analogues are introduced and we prove that for $ u \in {H^\ast}(X;\mathbb{Q})$, $ {\text{dcat}_0}\;u = 1 \Leftrightarrow {u^2} = 0$ and $ u$ is spherical.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1993-1112375-X
PII: S 0002-9947(1993)1112375-X
Keywords: Lusternik-Schnirelmann category, minimal models
Article copyright: © Copyright 1993 American Mathematical Society