Long exact sequence of a pair in singular cohomology

In every degree, restriction $r:C^n\to E^n$ is onto: extend a function on simplices in $A$ by zero on all other simplices of $X$, using [F2]. Its kernel is exactly $D^n$ by [F1], and $j:D^n\to C^n$ is inclusion. These maps commute with $\delta$ by [F1] and [F3]. Thus $0\to D\stackrel{j}{\to }C\stackrel{r}{\to }E\to 0$ is termwise exact. Extension by zero is a linear degreewise section, and is not asserted to be a cochain map. For negative degrees the exact row consists of zeros.

Let $a\in E^n$ be a cocycle and choose an extension $\tilde{a}$ as in step 1.1. Then $r\delta \tilde{a}=\delta a=0$, so $\delta \tilde{a}\in D^{n+1}$; it is closed by ${\delta }^2=0$. Two extensions differ by an element $d\in D^n$ and their differentials differ by the relative coboundary $\delta d$. If $a^{\mathrm{\prime }}=a+\delta b$, extend $b$ to $\tilde{b}$; then $\tilde{a}+\delta \tilde{b}$ extends $a^{\mathrm{\prime }}$ and has the same differential as $\tilde{a}$. Combining the two observations proves representative independence. Sum extensions and integer multiples prove that $\mathrm{\partial }[a]=[\delta \tilde{a}]$ is a homomorphism.

At $H^n(C)$, every relative class restricts to zero. Conversely if a cocycle $c$ has $[rc]=0$, write $rc=\delta b$ in $E^n$. Extend $b\in E^{n-1}$ to $\tilde{b}\in C^{n-1}$. Then $c-\delta \tilde{b}$ is a relative cocycle representing $[c]$. For $n=0$, $b$ and its extension are zero, since negative cochains vanish; the same reasoning is valid. Hence this kernel is exactly the relative image.

At $H^n(E)$, a global cocycle restricting to $a$ has zero connecting class. Conversely if $\mathrm{\partial }[a]=0$, an extension has $\delta \tilde{a}=\delta d$ for some $d\in D^n$. Then $\tilde{a}-d$ is a cocycle of $C$ restricting to $a$, so $[a]$ lies in the restriction image. This proves exactness there in both directions.

At $H^{n+1}(D)$, a connecting representative $\delta \tilde{a}$ is a coboundary in $C$, so its class maps to zero. Conversely if a relative cocycle $d$ is a global coboundary $d=\delta c$, then $rc$ is a cocycle of $E$ since $\delta rc=rd=0$. Its connector, using the extension $c$, is $[d]$. This proves exactness at the third type of position. If this relative degree is zero, $c\in C^{-1}=0$ forces $d=0$, establishing the initial injection explicitly.

All integer degrees and all three types of position are covered by steps 3.1, 2.2 and 3.2, proving the long exact sequence and its connector. If $A=\mathrm{\varnothing }$, then $D=C$ and $E=0$, so the sequence consists of identity and zero maps. If $A=X$, then $D=0$ and restriction is identity. Empty $X$ or zero $G$ give zero sequences. For a point and either of its two subspaces these same endpoint cases apply. The explicit extension-by-zero function is available without selecting any elements of $G$ beyond its specified zero; no AC, projectivity or injectivity of $G$ is used.