Module sectional category of products

Adapting a result of Félix–Halperin–Lemaire concerning the Lusternik–Schnirelmann category of products, we prove the additivity of a rational approximation for Schwarz’s sectional category with respect to products of certain fibrations.J.C. is supported by the Polish National Science Centre Grant 2016/21/ P/ST1/03460 within the European Unions Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie Grant Agreement No. 665778 and by the Belgian Interuniversity Attraction Pole (IAP) within the framework “Dynamics, Geometry and Statistical Physics” (DYGEST P7/18). L.V. is partially supported by Portuguese Funds through FCT – Fundação para a Ciência e a Tecnologia, within the Project UID/MAT/00013/2013

Secondary LS category of measured laminations

In the author's Ph.D., a version of the tangential LS category for foliated spaces depending on a transverse invariant measure, called the measured category, was introduced. Unfortunately, the measured category vanishes easily. When it is zero, the rate of convergence to zero of the quantity involved in the definition, by taking arbitrarily large homotopies, gives a new invariant, called the secondary measured category. Several versions of classical results are proved for the secondary measured category. It is also shown that the secondary measured category is a transverse invariant related to the growth of (pseudo)groups. The equality between secondary category and the growth of a group is done in the case of free suspensions by Rohlin groups.Comment: 14 pages. arXiv admin note: substantial text overlap with arXiv:1112.500

Measurable versions of the LS category on laminations

We give two new versions of the LS category for the set-up of measurable laminations defined by Berm\'udez. Both of these versions must be considered as "tangential categories". The first one, simply called (LS) category, is the direct analogue for measurable laminations of the tangential category of (topological) laminations introduced by Colman Vale and Mac\'ias Virg\'os. For the measurable lamination that underlies any lamination, our measurable tangential category is a lower bound of the tangential category. The second version, called the measured category, depends on the choice of a transverse invariant measure. We show that both of these "tangential categories" satisfy appropriate versions of some well known properties of the classical category: the homotopy invariance, a dimensional upper bound, a cohomological lower bound (cup length), and an upper bound given by the critical points of a smooth function.Comment: 22 page

Les premières étapes du calcul symbolique.

Lusternik L. A. Les premières étapes du calcul symbolique.. In: Revue d'histoire des sciences, tome 25, n°3, 1972. pp. 201-206