Transverse Lusternik–Schnirelmann category of foliated manifolds

Colman, Hellen; Macías–Virgós, E.

doi:10.1016/s0040-9383(99)00067-1

Cited by 16 publications

(29 citation statements)

References 13 publications

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“…When F is a compact Hausdorff foliation of a compact manifold, cat ∩ | (M ) is always finite [5,9]. Our main result, Theorem 6.1, gives estimates from above and below for cat ∩ | (M ), based on the geometry of the foliation and the topology of the exceptional set E. Our examples show that cat ∩ | (M ) can be arbitrarily large for a fixed quotient space M/F .…”

Section: Introductionmentioning

confidence: 99%

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LS-category of compact Hausdorff foliations

Colman

Hurder

2003

Trans. Amer. Math. Soc.

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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, . . . , ∞}. A foliation with all leaves compact and Hausdorff leaf space M/F is called compact Hausdorff. The transverse saturated category cat ∩ | 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 cat ∩ | (M ) in terms of the geometry of F and the Epstein filtration of the exceptional set E. The exceptional set is the closed saturated foliated space which is the union of the leaves with non-trivial holonomy. We prove thatWe 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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Section: Introductionmentioning

confidence: 99%

“…The transverse Lusternik-Schnirelmann category of a foliated manifold (M, F ) was introduced by Colman in [5,9,6]. The key concept is that an open set U ⊂ M is transversely categorical if the inclusion factors through a leaf, up to foliated homotopy.…”

Section: Introductionmentioning

confidence: 99%

LS-category of compact Hausdorff foliations

Colman

Hurder

2003

Trans. Amer. Math. Soc.

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“…If the orbifold is effective, the LS-category for orbifolds as groupoids coincide with the LS-category for orbifolds as defined by the author in [9] using local charts.…”

Section: The Lusternik-schnirelmann Category Of a Lie Groupoid 5555mentioning

confidence: 99%

“…James and J.R. Morris [21], the equivariant category, by E. Fadell [16] and the transverse and tangential categories for foliated manifolds [12,11,8].…”

Section: Introductionmentioning

confidence: 99%

The Lusternik-Schnirelmann category of a Lie groupoid

Colman

2010

Trans. Amer. Math. Soc.

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Abstract. We propose a new homotopy invariant for Lie groupoids which generalizes the classical Lusternik-Schnirelmann category for topological spaces. We use a bicategorical approach to develop a notion of contraction in this context. We propose a notion of homotopy between generalized maps given by the 2-arrows in a certain bicategory of fractions. This notion is invariant under Morita equivalence. Thus, when the groupoid defines an orbifold, we have a well-defined LS-category for orbifolds. We prove an orbifold version of the classical Lusternik-Schnirelmann theorem for critical points.

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“…In the case where all leaves of F are compact, Colman [12,18] proved a lower bound estimate for the number of critical leaves in terms of the transverse LS category cat ∩ | (M, F), which the estimate (3) generalizes to the general case when F has non-compact leaves. The transverse LS category cat ∩ | (M, F) is infinite when F has no compact leaves, while the essential transverse LS category cat We say that the image leaf L 1 is foliated isotopic to L.…”

Section: Introductionmentioning

confidence: 99%