Relative homotopy group (${\pi }_r(X,A)$)

Based relative maps of a disk into $X$ with its boundary mapped into $A$, modulo homotopies preserving these conditions. Replacing a map by its mapping cylinder gives relative groups for a general map. They fit into the long exact sequence of relative homotopy groups and detect connectivity of the map.

Inclusion, the map sending an absolute disk class to a relative one, and restriction to the disk boundary give this exact sequence. Its low-dimensional end is interpreted with pointed sets and the usual fundamental-group action. For a mapping-cylinder pair it measures the failure of a map to induce homotopy isomorphisms.