Paths in space. In this chapter we shall discuss path groups with the aim of illustrating how the definition of groups by generators and relations arises in a natural manner from topological problems. The presentation of the concepts associated with path groups will lean heavily on the reader's space intuition.

We shall consider closed paths that begin and end at a fixed point P (the “origin”) in space. Notice that we use the designation “path” rather than “curve” to emphasize that we are concerned with a definite direction along the path. This is in keeping with our treatment of paths along directed segments of the graph of a group. We shall not be concerned with the shape of a path. On the contrary, we shall be interested in the possible effects of changing the shape of a path. We shall call two paths a1 and a2 through P “equal” or “the same path” if we can deform a1 into a2 by a continuous change. We have already described such paths as “topologically equivalent” (see p. 52). Another term for denoting such equality of paths is “homotopy”; and the “equal” paths a1 and a2 are said to be homotopic.

It might appear, at first sight, that all closed paths through P are equal, or homotopic. If we take a point P in “empty” space, then any closed path a through P can be continuously shrunk to the point P.