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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48 .

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Transverse LS category for Riemannian foliations
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by Steven Hurder and Dirk Töben PDF
Trans. Amer. Math. Soc. 361 (2009), 5647-5680 Request permission


We study the transverse Lusternik-Schnirelmann category theory of a Riemannian foliation $\mathcal {F}$ on a closed manifold $M$. The essential transverse category $\operatorname {cat}^e_{\cap {\mkern -9mu}\mid }(M,\mathcal {F})$ is introduced in this paper, and we prove that $\operatorname {cat}^e_{\cap {\mkern -9mu}\mid }(M,\mathcal {F})$ is always finite for a Riemannian foliation. Necessary and sufficient conditions are derived for when the usual transverse category $\operatorname {cat}_{\cap {\mkern -9mu}\mid }(M,\mathcal {F})$ is finite, and thus $\operatorname {cat}^e_{\cap {\mkern -9mu}\mid }(M,\mathcal {F}) = \operatorname {cat}_{\cap {\mkern -9mu}\mid }(M,\mathcal {F})$ 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 $\operatorname {cat}^e_{\cap {\mkern -9mu}\mid }(M,\mathcal {F})$ 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 $\operatorname {cat}^e_{\cap {\mkern -9mu}\mid }(M,\mathcal {F}) = \operatorname {cat}_{\mathbf O(q)}(\widehat W)$. 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 $\operatorname {cat}^e_{\cap {\mkern -9mu}\mid }(M,\mathcal {F})$ 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
  • MR Author ID: 90090
  • ORCID: 0000-0001-7030-4542
  • Email:
  • Dirk Töben
  • Affiliation: Mathematisches Institut, Universität zu Köln, Weyertal 86-90, 50931 Köln, Germany
  • Email:
  • 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
  • © Copyright 2009 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 361 (2009), 5647-5680
  • MSC (2000): Primary 57R30, 53C12, 55M30; Secondary 57S15
  • DOI:
  • MathSciNet review: 2529908