Given two connected subsets Y X of the set of the surfels of a connected digital surface, we propose three equivalent ways to express that Y is homotopic to X. The rst characterization is based on sequential deletion of simple surfels. This characterization enables us to deene thinning algorithms within a digital Jordan surface. The second characterization is based on the Euler characteristics of sets of surfels. This characterization enables us, given two connected sets Y X of surfels, to decide whether Y is nhomotopic to X. The third characterization is based on the (digital) fundamental group.