The simplicial Lusternik-Schnirelmann category

Specialeforsvar: Erica Muniz

Titel: The simplicial Lusternik-Schnirelmann category

Resume: In the present thesis we describe the simplicial Lusternik-Schnirelmann category and the simplicial geometric category for finite simplicial complexes. Moreover, we relate these categories with the Lusternik-Schnirelmann and geometric categories of finite T0-spaces, showing that they behave in a similar way under specific conditions. For example, the L-S category is a homotopy invariant and the simplicial L-S category is a strong homotopy invariant, and the geometric category for finite T0-spaces increases under elimination of beat points and its simplicial version increases under elimination of dominated vertices. With this aim, we describe the structure of finite T0-spaces and finite simplicial complexes and we relate them through the Order Complex functor and the Face Poset functor. Finally, we show that the L-S category of the geometric realisation provides a lower bound for the simplicial L-S category and a lower bound for the simplicial L-S category of the iterated subdivision. There will be computations of simplicial L-S categories and examples of the main theorems. However, no examples have been found for simplicial complexes, such that the value of the geometric category strictly increases under removing a dominated vertex, or such that the category of its face poset is strictly smaller than its simplicial category. Moreover, there is no criteria for defining an upper bound for the value of the simplicial L-S category.

Vejleder: Jesper Michael Møller
Censor: Steen Markvorsen