Equivariant LS-Category and Equivariant Topological Complexity

Abstract

In this thesis we consider topological spaces endowed with an action of a topological group, and we develop a new concept to study these spaces. This concept is called orbit class and is often a good replacement for the well-known concept or- bit type. Using the concept of orbit class, we de ne a partial ordering on the set of all orbit classes. This partial order not only gives a partition on the topological space based on the orbits, but it also gives a discrete combinatorial translation of the topological space. We also use the properties of the orbit class to study equivariant LS-category and equivariant topological complexity. Equivariant LS-category was introduced by Marzantowicz in 1989, as a generalization of LS-category. Since then, equivariant LS-category has been studied by mathematicians and many results with di erent conditions have been developed. Equivariant topological complexity was introduced by Colman and Grant in 2012, as a generalization of topological complexity. In 2015, Lubawski and Marzantowicz introduced the invariant topological complexity as another generalization of the topological complexity and they claimed that their proposed invariant is more e cient than the equivariant topological complexity. In this thesis we study the equivariant LS-category and give some new results found by applying the properties of orbit class. We also study both the equivariant topological complexity and the invariant topological complexity. By using results from orbit class we show that in most cases the invariant topological complexity is in nite. In particular, if a topological space has more than one minimal orbit class then the invariant topological complexity is in nite. Finally, we study some particular cases of locally standard torus manifolds, and calculate their LS-category, topological complexity, equivariant LS-category, and invariant topological complexity. We also give counterexamples to two theorems from a published paper by Colman and Grant [10], and prove a modi ed version of one of those theorems.