Homotopy Path Spaces for Families of Admissible Paths

Abstract

Motivated by the search for universal orbits in geometric control theory and universal topological semigroups over local semigroups in topological groups, particularly Lie groups, we introduce a very general axiomatic notion of "trajectory," namely a topological space equipped with a family of paths satisfying certain properties that make it what we call an "admissible family." Considering one-parameter families of these paths as a homotopy allows us to construct a homotopy path space, our main object of consideration. We identify a distinguished subfamily of admissible paths via what we call a selection or partial selection and show how this insures collapses in the homotopy path space structure. We establish sufficient conditions to identify the homotopy path space as the simply connected covering or as residing in the simply connected covering and point out certain of its universal properties in this setting. We close by applying and illustrating our results in the setting of Lie groups and semigroups.