Today, we will define *relative* versions of the homotopy groups, and show that they fit into an exact sequence. So let be a pointed space and a subspace containing the basepoint. Let . Then we define

to be the pointed homotopy class of maps . Here has a basepoint, which is located on the boundary .

Definition 1is called the -threlative homotopy groupof the pair . (We have not yet shown that it is a group.)

Another perhaps more geometric way of thinking of the relative homotopy groups is as follows. Namely, it is the homotopy class of maps , where is the -cube and is the complement of the front face . So on the boundary, such a map is except possibly on the front face, where it is at least in . The reason is that if we quotient by , we get the pair . (more…)