Homotopy between constant loop and composite of loop with inverse
Suppose is a point in a topological space . Suppose is a loop based at , i.e., is a continuous map from the closed unit interval to such that . Denote by the loop defined as . Denote by the constant loop at the point .
Denote by the composition of loops by concatenation. Then, the loops and are both homotopic to .
Note that since , it suffices to show that is homotopic to .
We note that , so we have:
The map to which we want to homotope this is:
The homotopy is given by:
Here is a pictorial description of the homotopy:
Then, the maps and are homotopic maps.