Long exact sequence of relative homotopy groups ($\cdots \to {\pi }_r(A)\to {\pi }_r(X)\to {\pi }_r(X,A)\to {\pi }_{r-1}(A)\to \cdots$)

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.