The equivalence relation induced by homotopy starts to enter the realm of algebraic topology, which is a branch of mathematics that characterizes the structure of topological spaces in terms of algebraic objects, such as groups. These resulting groups have important implications for motion planning. Therefore, we give a brief overview. First, the notion of a group must be precisely defined. A group is a set, , together with a binary operation, , such that the following group axioms are satisfied:
An important property, which only some groups possess, is commutativity: for any . The group in this case is called commutative or Abelian. We will encounter examples of both kinds of groups, both commutative and noncommutative. An example of a commutative group is vector addition over . The set of all 3D rotations is an example of a noncommutative group.
Steven M LaValle 2012-04-20