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, $ G$, together with a binary operation, $ \circ$, such that the following group axioms are satisfied:

  1. (Closure) For any $ a, b \in G$, the product $ a \circ b \in
  2. (Associativity) For all $ a, b, c \in G$, $ (a \circ b) \circ
c = a \circ (b \circ c)$. Hence, parentheses are not needed, and the product may be written as $ a \circ b \circ c$.
  3. (Identity) There is an element $ e \in G$, called the identity, such that for all $ a \in G$, $ e \circ a = a$ and $ a \circ e
= a$.
  4. (Inverse) For every element $ a \in G$, there is an element $ a^{-1}$, called the inverse of $ a$, for which $ a \circ a^{-1} =
e$ and $ a^{-1} \circ a = e$.
Here are some examples.

Example 4..7 (Simple Examples of Groups)   The set of integers $ {\mathbb{Z}}$ is a group with respect to addition. The identity is 0, and the inverse of each $ i$ is $ -i$. The set $ {\mathbb{Q}}
\setminus 0$ of rational numbers with 0 removed is a group with respect to multiplication. The identity is $ 1$, and the inverse of every element, $ q$, is $ 1/q$ (0 was removed to avoid division by zero). $ \blacksquare$

An important property, which only some groups possess, is commutativity: $ a \circ b = b \circ a$ for any $ a, b \in G$. 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 $ {\mathbb{R}}^n$. The set of all 3D rotations is an example of a noncommutative group.

Steven M LaValle 2012-04-20