STRING TOPOLOGICAL ROBOTICS

摘要

We claim here to link two well known theories; namely the string topology (founded by Chas and Sullivan in [3]) and the topological robotics (founded by Farber some few years later, in [7]). For our purpose, we consider G a compact Lie group acting transitively on a path connected n-manifold X. On the set M~(LP)(X) of the so-called loop motion planning algorithms, we define and discuss the notion of transversality. Firstly, we define an intersection loop motion planning product at level of chains of M~(LP)(X). Secondly, we define a boundary operator on the chains of M~(LP)(X) and extend this intersection product at level of homology to a string loop motion planning product. Finally, we show that this string product induces on the shifted string loop motion planning homology H*(M~(LP) (X)):= H_(*+2n)(M~(LP)(X)) a structure of an associative and commutative graded algebra (acga). By the end, we ask how one may extend this acga-structure to a structure of Gerstenhaber algebra or that of a Batalin-Vilkovisky algebra. Some ideas will be suggested.