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

 

LS-category of compact Hausdorff foliations


Authors: Hellen Colman and Steven Hurder
Journal: Trans. Amer. Math. Soc. 356 (2004), 1463-1487
MSC (2000): Primary 55M30, 57R30; Secondary 57S05, 57N80
Published electronically: November 4, 2003
MathSciNet review: 2034314
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Abstract: The transverse (saturated) Lusternik-Schnirelmann category of foliations, introduced by the first author, is an invariant of foliated homotopy type with values in $\{1,2, \ldots, \infty\}$. A foliation with all leaves compact and Hausdorff leaf space $M/\mathcal{F}$ is called compact Hausdorff. The transverse saturated category $\operatorname{cat}_{\mathbin{\cap{\mkern-9mu}\mid}\,\,}M$ of a compact Hausdorff foliation is always finite.

In this paper we study the transverse category of compact Hausdorff foliations. Our main result provides upper and lower bounds on the transverse category $\operatorname{cat}_{\mathbin{\cap{\mkern-9mu}\mid}\,\,}(M)$ in terms of the geometry of $\mathcal{F}$ and the Epstein filtration of the exceptional set $\mathcal{E}$. The exceptional set is the closed saturated foliated space which is the union of the leaves with non-trivial holonomy. We prove that

\begin{displaymath}\max \{\operatorname{cat}(M/{\mathcal{F}}), \operatorname{ca... ...me{cat}_{\mathbin{\cap{\mkern-9mu}\mid}\,\,}(\mathcal{E}) + q.\end{displaymath}

We give examples to show that both the upper and lower bounds are realized, so the estimate is sharp. We also construct a family of examples for which the transverse category for a compact Hausdorff foliation can be arbitrarily large, though the category of the leaf spaces is constant.


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Additional Information

Hellen Colman
Affiliation: Department of Mathematics, University of Illinois at Chicago, 322 SEO (M/C 249), 851 S. Morgan Street, Chicago, Illinois 60607-7045
Email: hcolman@math.uic.edu

Steven Hurder
Affiliation: Department of Mathematics, University of Illinois at Chicago, 322 SEO (M/C 249), 851 S. Morgan Street, Chicago, Illinois 60607-7045
Email: hurder@uic.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-03-03459-7
PII: S 0002-9947(03)03459-7
Received by editor(s): August 1, 2002
Published electronically: November 4, 2003
Additional Notes: The first author was partially supported by grants from the Xunta Galicia, Spain, and University of Sheffield, EU RTN1-1999-00176, Modern Homotopy Theory
The second author was partially supported by NSF Grant DMS-9704768
Article copyright: © Copyright 2003 American Mathematical Society