Long exact sequence of relative homotopy groups

At ${\pi }_n(X)$, an absolute class represented by a map into A is relatively null: in the disk model contract its domain to the marked boundary point, through maps into A. Conversely a class killed by j compresses into A with its full boundary fixed by F2; since that boundary was constant x0, the compressed representative defines an absolute class in ${\pi }_n(A)$ mapping to the original. This proves both image inclusions for all n≥1.

At ${\pi }_n(X,A)$ for n≥2, the boundary of an absolute representative is constant, so $\mathrm{\partial }{j}_{\ast }=0$. If $a(u,t)$ has distinguished face $h(u)=a(u,0)$ nullhomotopic in A, take a boundary-fixed $B(u,s)$ from h to x0. For a collar width $0<\lambda \le 1$ define $a_{\lambda }(u,t)=B(u,\lambda -2t)$ for $0\le t\le \lambda \mathrm{/}2$, and $a(u,(t-\lambda \mathrm{/}2)\mathrm{/}(1-\lambda \mathrm{/}2))$ for $\lambda \mathrm{/}2\le t\le 1$. At the seam both values are h. As λ goes from 0 to 1, using this formula for $0<\lambda \le 1$ and $a_0=a$, it gives a relative homotopy: near λ=t=0 both arguments tend to the common value h. At λ=1 the bottom value is $B(u,1)=x_0$, and all other faces are fixed, so it is an absolute representative.

At ${\pi }_n(A)$ for n≥1, a relative $(n+1)$-cube is itself an X-nullhomotopy of its distinguished face, so $i_{\ast }\mathrm{\partial }=0$. Conversely, if an A-based cube is nullhomotopic in X rel its boundary, that nullhomotopy, with time as the final coordinate, is a relative $(n+1)$-cube whose boundary is the given cube. This proves equality of kernel and image there.

At ${\pi }_1(X,A)$, a path α starts at some $a\in A$ and ends at x0. Its boundary component is distinguished precisely when a can be joined to x0 in A. Given a path $p:x_0\to a$ in A, the based loop $p\ast \alpha$ has the same relative class as α: attach the terminal segment $p{\mathrm{\mid }}_{[1-s,1]}$ ahead of α with width s/2. At s=0 it is α; at s=1 it is the loop, the changing initial endpoint stays in A, and the seam agrees at a. Conversely the initial point of a based loop is x0, and any relative homotopy keeps its initial endpoint within the same A-component.

At ${\pi }_0(A)$, the component of a maps to the distinguished X-component exactly when a path in X joins a to x0. Such a path is a relative degree-one representative with boundary component [a]. Conversely every relative path provides that connection. Components of X not meeting A are not constrained by this calculation.

Composition with a map of based pairs commutes pointwise with inclusion and distinguished-face restriction. Hence every square of the displayed sequence commutes by F1 and F3. Steps 1.1–1.5 establish exactness at all eligible terms, with the pointed-set interpretation in the low tail.