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Journal of the London Mathematical Society 2002 65(3):745-756; doi:10.1112/S0024610702003113
© 2002 by London Mathematical Society
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© The London Mathematical Society

Tangential Lusternik–Schnirelmann Category of Foliations

Hellen Colman and Enrique Macias-Virgós

Department of Mathematics (m/c 249), University of Illinois 851 S Morgan Street, Chicago, IL 60607-7045, USA, hcolman{at}math.uic.edu
Departamento de Xeometria e Topoloxia, Facultade de Matemáticas, Universidade de Santiago de Compostela 15706 – Galicia, Spain, macias{at}zmat.usc.es

Received 8 March 2001. Revision received 30 September 2001.

A new invariant of (integrable) homotopy type for foliations is introduced: the tangential category of a foliated manifold.

The classical Lusternik–Schnirelmann theory is generalized to foliations and the relations of the tangential category with other known invariants such as the fibrewise and the equivariant category are studied. Cohomological lower bounds are provided in terms of foliated cohomology.

If the foliation is a product, the tangential category coincides with the ordinary category of the leaves. In general it is just bounded below. Estimates are given of the tangential category for compact-Hausdorff foliations and suspensions. Examples show that the lower and upper bounds are realized.


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