LS category, foliated spaces and transverse invariant measure

Abstract

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 di↵erent areas like robotics. Many variants of the LS category has been given; in particular, E. Mac´ıas 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.