The Inverse Path and the constant path:

Suppose $\gamma: [0,1] \rightarrow X$ is a path then the inverse path $\gamma^{-1}(t)$ is the path traced in the reversed direction namely the map $\gamma^{-1} : [0,1] \rightarrow X$ given by

$$\gamma^{-1}(t)=\gamma(1-t).$$

The initial point of $\gamma$ is the terminal point of $\gamma^{-1}$ and vice versa.

The constant path at $x_0$ is the path $\varepsilon_{x_0}: [0,1] \longrightarrow X$ given by

$$\varepsilon_{x_0}(t)=x_0$$    for all $$t\in [0,1].$$

The following lemma summarizes the main properties of the constant and the inverse paths in terms of the homotopy classes of paths. Theorem (7.6) spells out the associativity of multiplication of homotopy classes of paths. The reader would see analogies with the defining properties of a group.