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Transactions of the American Mathematical Society

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



Transverse LS category for Riemannian foliations

Authors: Steven Hurder and Dirk Töben
Journal: Trans. Amer. Math. Soc. 361 (2009), 5647-5680
MSC (2000): Primary 57R30, 53C12, 55M30; Secondary 57S15
Published electronically: June 16, 2009
MathSciNet review: 2529908
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Abstract: We study the transverse Lusternik-Schnirelmann category theory of a Riemannian foliation $ \mathcal{F}$ on a closed manifold $ M$. The essential transverse category $ \operatorname{cat}^e_{\mathbin{\cap{\mkern-9mu}\mid} }(M,\mathcal{F})$ is introduced in this paper, and we prove that $ \operatorname{cat}^e_{\mathbin{\cap{\mkern-9mu}\mid} }(M,\mathcal{F})$ is always finite for a Riemannian foliation. Necessary and sufficient conditions are derived for when the usual transverse category $ \operatorname{cat}_{\mathbin{\cap{\mkern-9mu}\mid} }(M,\mathcal{F})$ is finite, and thus $ \operatorname{cat}^e_{\mathbin{\cap{\mkern-9mu}\mid} }(M,\mathcal{F}) = \operatorname{cat}_{\mathbin{\cap{\mkern-9mu}\mid} }(M,\mathcal{F})$ holds.

A fundamental point of this paper is to use properties of Riemannian submersions and the Molino Structure Theory for Riemannian foliations to transform the calculation of $ \operatorname{cat}^e_{\mathbin{\cap{\mkern-9mu}\mid} }(M,\mathcal{F})$ into a standard problem about $ \mathbf O(q)$-equivariant LS category theory. A main result, Theorem 1.6, states that for an associated $ \mathbf O(q)$-manifold $ \widehat W$, we have that $ \operatorname{cat}^e_{\mathbin{\cap{\mkern-9mu}\mid} }(M,\mathcal{F}) = \operatorname{cat}_{\mathbf O(q)}(\widehat W)$. Hence, the traditional techniques developed for the study of smooth compact Lie group actions can be effectively employed for the study of the LS category of Riemannian foliations.

A generalization of the Lusternik-Schnirelmann theorem is derived: given a $ C^1$-function $ f \colon M \to \mathbb{R}$ which is constant along the leaves of a Riemannian foliation $ \mathcal{F}$, the essential transverse category $ \operatorname{cat}^e_{\mathbin{\cap{\mkern-9mu}\mid} }(M,\mathcal{F})$ is a lower bound for the number of critical leaf closures of $ f$.

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Additional Information

Steven Hurder
Affiliation: Department of Mathematics, University of Illinois at Chicago, 322 SEO (m/c 249), 851 S. Morgan Street, Chicago, Illinois 60607-7045

Dirk Töben
Affiliation: Mathematisches Institut, Universität zu Köln, Weyertal 86-90, 50931 Köln, Germany

Keywords: Riemannian foliation, Lusternik-Schnirelmann category, Riemannian submersion, compact Hausdorff foliation, Epstein filtration
Received by editor(s): October 4, 2006
Published electronically: June 16, 2009
Additional Notes: The first author was supported in part by NSF grant DMS-0406254
The second author was supported by the Schwerpunktprogramm SPP 1154 of the DFG
Article copyright: © Copyright 2009 American Mathematical Society