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…)