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Definition: Let F: IxI -> X be a continuous map such that F(0, t) = F(1, t) for all t. Then for each t, the map f_t(S_ = F(s, t) is a loop in X. The map F is called a free homotopy between the loops f_0 and f_1. It is a homotopy of loops in which the base point of the loop is allowed to move during the homotopy.
Source: Topology (second edition) by James R. Munkres