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:

- (
**Closure**) For any , the product . - (
**Associativity**) For all , . Hence, parentheses are not needed, and the product may be written as . - (
**Identity**) There is an element , called the*identity*, such that for all , and . - (
**Inverse**) For every element , there is an element , called the*inverse*of , for which and .

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