Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Lower bounds for the L.-S. category of products

Authors: Barry Jessup and Bitjong Ndombol
Journal: Proc. Amer. Math. Soc. 117 (1993), 839-842
MSC: Primary 55P50; Secondary 55M30, 55P60, 55P62
MathSciNet review: 1152985
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Abstract: Halperin and Lemaire introduced L.-S. category type invariants $ {\text{left-}}\operatorname{Mcat} (A)$ and $ {\text{right-}}\operatorname{Mcat} (A)$(/l) for certain differential algebras $ (A,d)$. In particular, they proved that if $ (A,d) = {C^{\ast}}(S,k)$ is the $ k$-valued singular cochains on $ 1$-connected space $ S$, then these invariants are lower bounds for the classical category $ {\text{cat}}(S)$. We use an explicit model for Ganea's space due to Felix, Halperin, Lemaire, and Thomas to prove $ \operatorname{lMcat} (A \otimes B) \leqslant \operatorname{lMcat} (A) + e(B)$, over any field, where $ e$ denotes Toomer's invariant. This proves Ganea's conjecture for Mcat over fields of arbitrary characteristic.

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Keywords: Lusternik-Schnirelmann category, free tensor model, Ganea's conjecture
Article copyright: © Copyright 1993 American Mathematical Society