PlanetPhysics/Thin Double Track

Let us consider a square map $$ u:I^{2} \longrightarrow X $$ in a topological space $$ X $$. Then the homotopy class of $$ u $$ relative to the boundary $$ \partial{I}^{2} $$ of $$ I $$ is called a {\it double track}. A double track is {\it thin} if it has a thin representative.