# Categoria ls, espacios foliados y medida transversa invariante

- Carlos Meniño Coton

- Jesús Antonio Álvarez López Director

Universidade de defensa: Universidade de Santiago de Compostela

Ano de defensa: 2012

- Enrique Macías-Virgós Presidente
- Fernando Alcalde Cuesta Secretario
- Alvaro Lozano Rojo Vogal
- Gilbert Hector Vogal
- Elmar Vogt Vogal

Tipo: Tese

## Resumo

The LS category is a homotopy invariant of topological spaces introduced by Lusternik and Schnirelmann in 1934, which was originally motivated by problems of variational calculus. It is defined as the minimum number of contractible open subsets needed to cover a space. Besides its original variational application, it became an important tool in homotopy theory, and it was applied in other different areas like robotics. Many variants of the LS category has been given; in particular, E.~Macías and H.~Colman introduced a tangential version for foliations, where they used leafwise contractions to transversals. In this thesis, the following new versions of the tangential LS category are introduced: - Measurable category. It is the direct adaptation of the tangential LS category to measurable laminations, where we have an ambient measure space with a leaf topology (the ambient space topology is missing). Then it is natural to define a measurable category as the minimum number of measurable (leafwise) open sets contractible to transversals by measurable (leafwise) continuous deformations. This version is relevant because, by using the obvious forgetful functor from laminations to measurable laminations, it provides a lower bound of the tangential LS category, which is easier to deal with (it is easier to handle measurable functions than continuous ones). - Measurable Lambda-category. This version is also defined for measurable laminations, but now a transverse invariant measure Lambda is used to "count" the tangential deformations of measurable open sets to transversals; i.e., it is the infimum of the sums of the Lambda-measures of the transversals resulting from tangential deformations of the sets of a measurable open covering. - Topological Lambda-category. This version is defined for laminations; i.e., there is also an ambient space topology. Then its definition is like the measurable Lambda-category, depending on a transverse invariant measure Lambda, but now we take coverings that are open on the ambient space, as well as tangential deformations that are continuous on the ambient space. - Secondary Lambda-category. The computations in examples show that, unfortunately, the topological Lambda-category vanishes very easily (e.g. when the leaves are dense). For that reason, when the topological Lambda-category vanishes, an induced secondary invariant is defined as the rate of convergence to zero of the expression giving the Lambda-category when the "length" of the tangential deformations increases to infinity. - Dynamical category. It is defined like the topological Lambda-category, but, instead of using a transverse invariant measure, the diameter given by an ambient metric is used to "measure" the size of the transversals resulting from tangential deformations. The main advantage of this definition is that it does not require any transverse invariant measure, whose existence is a very restrictive condition. In the case of smooth foliations on closed Riemannian manifolds, the positivity or nullity of the dynamical category depends only on the foliation. - Secondary dynamical category. It is the secondary invariant associated to the dynamical category in the same way as the secondary Lambda-category is associated to the topological Lambda-category. The main achievements of the thesis are: - preliminary studies of transverse invariant measures and measurable cohomology, - computations of these variations of the tangential category in examples, - corresponding versions of the main theorems about the classical LS category, and - new theorems that are special of the lamination setting. More precisely, the following kind of results are given: - Extension of transverse invariant measures. It is proved that any transverse invariant measure extends to a measure on the ambient space, which is unique if some condition of coherency is assumed. This extension can be considered as a pairing of the given transverse invariant measure with the counting measure on the leaves. - Measurable cohomology. This is a new version of cohomology introduced for measurable laminations. The definition involves tangential cochains which give measurable functions when applied to transversely measurable families of simplices. Some standard properties of cohomology are extended to this setting, and it is computed in some examples. - Examples. Expressions of the above versions of the tangential category are given in the cases of compact leaves, dense leaves or suspensions with Rohlin groups. - Homotopy invariance. It is proved that these versions of the tangential LS category are invariant by tangential homotopy equivalences compatible with the structure involved in their definitions (transverse invariant measures or Lipschitz types of metrics). - Transverse invariance. The positivity or nullity of the above primary categories, and the corresponding secondary categories as well, are shown to depend only on the transverse holonomy pseudogroup. For this purpose, corresponding versions of primary and secondary LS categories are defined for pseudogroups, and they turn out to be invariant by pseudogroup equivalences. - Growth of pseudogroups. We prove that the secondary categories of a pseudogroup are related with its growth. - Dimensional upper bound. The dimension of a manifold is an upper bound of its LS category. This was generalized to the tangential category by Singhof-Vogt. We also prove corresponding versions for our primary categories. - Cohomological lower bound. The usual lower bound of the LS category by the cup length is extended to our new versions of the tangential LS category. The secondary tangential categories require corresponding secondary versions of the cup length. - Semicontinuity. Singhof-Vogt have proved that the tangential LS category is upper semicontinous on the space of foliations on a fixed manifold. By using the same methods, we have also proved upper semicontinuity for our primary tangential categories, and certain lower semi-continuity for the secondary ones. - Critical points. For smooth functions on manifolds, one of the first theorems on the classical LS category states that it is a lower bound of the number of critical points, establishing its connection with variational calculus. That kind of result was missing even for the original tangential LS category. A version of that relation with critical points was proved for all versions of the tangential categories, including the original one of Colman-Macías. This could be considered as the most important result of the thesis. We use critical sets instead of critical points, which are "measured" with the tools used in the tangential categories (counting, measures or diameters). Secondary approaches to the critical sets are also considered in the case of secondary tangential categories. For the case of smooth functions on manifolds, the use of critical sets improves the classical result, showing the relevance of this point of view. For these kind of theorems, we have used Hilbert laminations with the purpose of applying them to "foliation variational calculus" in the future.