Long exact sequence of relative homotopy groups
= Long exact sequence of relative homotopy groups
{title2=$\cdots\to\pi_r(A)\to\pi_r(X)\to\pi_r(X,A)\to\pi_{r-1}(A)\to\cdots$}
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.