Today, we will define relative versions of the homotopy groups, and show that they fit into an exact sequence. So let {X} be a pointed space and {A \subset X} a subspace containing the basepoint. Let {n \geq 1}. Then we define

\displaystyle  \pi_n(X, A)

to be the pointed homotopy class of maps {(D^n, S^{n-1}) \rightarrow (X, A)}. Here {D^n} has a basepoint, which is located on the boundary {S^{n-1}}.

Definition 1 {\pi_n(X, A)} is called the {n}-th relative homotopy group of the pair {(X, A)}. (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 {(I^n, I^{n-1}, J^n) \rightarrow (X, A, x_0)}, where {I^n} is the {n}-cube and {J^n} is the complement of the front face {I^{n-1}}. So on the boundary, such a map is {x_0} except possibly on the front face, where it is at least in {A}. The reason is that if we quotient by {J^{n-1}}, we get the pair {(D^n, S^{n-1})}. (more…)