Free homotopy
= Free homotopy
A <free homotopy> between maps is an ordinary <homotopy> with no specified basepoint held fixed. Two loops at the same basepoint are freely homotopic precisely when their elements of the <fundamental group> are conjugate: a <homotopy> traces a connecting basepoint path, and the boundary of its parameter square gives conjugation. Conversely that connecting path can be slid around the loop to realize the conjugation.