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Let $(A, X)$ be a [[topological pair]]. That is, $X$ is a [[topological space]] and $A \subset X$ is a [[subspace topology|subspace]]. Write $\iota:A \hookrightarrow X$ for the [[inclusion map]].
> [!definition] Definition. ([[relative singular homology]])
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> [[singular (co)chain map and homomorphism induced by a continuous map|The map]] $\iota_{n}:C_{n}(A) \hookrightarrow C_{n}(X)$ is evidently [[injection|injective]], giving a left-[[exact sequence]] $0 \to C_{\bullet}(A) \to C_{\bullet}(X)$. The $i$th [[relative homology of an embedding of chain complexes|relative homology]] of this embedding of [[chain complex of modules|chain complexes]] is called the **$i$th relative (singular) homology of the pair $(X,A)$** and denoted $H_{i}(X,A)$.
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$H_{i}(-, -)$ is a [[covariant functor]] $\mathsf{TopPairs} \to \mathsf{Ab}$, obtained by composing the [[singular map of pairs functor]] $\mathsf{TopPairs} \to \mathsf{Chain Pairs}$ with the general [[relative homology of an embedding of chain complexes|relative homology functor]] $\mathsf{ChainPairs} \to \mathsf{Ab}$. [[relative homology of an embedding of chain complexes|Explicitly]], the functor $H_{i}(-,-)$ sends a [[topological pair|map of pairs]] $(X,A) \xrightarrow{f}(Y,B)$ to the [[linear map|homomorphism]] $\begin{align}
f_{*}:H_{i}(X,A) &\to H_{i}(Y,B) \\
\big[ [x] \big] & \mapsto \big[ [f_{i}(x)] \big].
\end{align}$
- [ ] relative singular cohomology and its (contravariant!) functor
> [!definition] Definition. ([[relative singular homology]] — unpacked)
> Here is an 'unpacked' version of the above definition. [[singular (co)chain map and homomorphism induced by a continuous map|The map]] $\iota_{n}:C_{n}(A) \hookrightarrow C_{n}(X)$ is evidently [[injection|injective]], giving a [[short exact sequence]] $0 \to C_{n}(A) \xhookrightarrow{\iota_{n}}C_{n}(X) \twoheadrightarrow \underbrace{ \frac{C_{n}(X)}{C_{n}(A)} }_{ := C_{n}(X, A) } \to 0.$
> The [[singular homology|differential]] $d_{n}:C_{n}(X) \to C_{n-1}(X)$ restricts to a map $C_{n}(A) \to C_{n-1}(A)$, and thus descends[^1] to a [[well-defined]] differential $d_{n}=\overline{d_{n}}:C_{n}(X,A) \to C_{n-1}(X,A)$ sending $[c] \mapsto [d_{n}(c)]$.
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> ```tikz
> \usepackage{tikz-cd}
> \usepackage{amsmath}
> \footnotesize
> \begin{document}
> % https://tikzcd.yichuanshen.de/#N4Igdg9gJgpgziAXAbVABwnAlgFyxMJZABgBpiBdUkANwEMAbAVxiRAGEB9MACgA0AlCAC+pdJlz5CKAIzkqtRizZdgYALQzh-IaPHY8BIgCZ51es1aIOnNZu19SAQQEBeADruAZgCc6AY2BVDS0dYSDbEO0XYRExEAwDKSIyGQULZWtPXwCItQcBcODowtcuXkcXEQUYKABzeCJQXwgAWyQAZmocCCRTRUs2TxgADyw4HDgAAgBCKc8IGhgfBiwwGGAobljqBjoAIxgGAAUJQ2kQHyw6gAscOOafNqQyEB6kOQHMkC3CPRAWu1EJ93oh+jcYHQoGwcAB3CAQqEIf6Al7dXqILogRHQ6xwhGQqDIijCIA
> \begin{tikzcd}
> C_n(X) \arrow[r, "d_n"] \arrow[d, two heads] & C_{n-1}(X) \arrow[r, two heads] & {C_{n-1}(X,A)=\frac{C_{n-1}(X)}{C_{n-1}(A)}} \\
> {\frac{C_{n}(X)}{C_{n}(A)}=C_n(X,A)} \arrow[rru, "\exists ! \overline{d_n}"'] & &
> \end{tikzcd}
> \end{document}
> ```
>
> The $i$th [[(co)homology of a complex|homology]] $H_{i}\big(C_{\bullet}(X,A)\big)$ of the [[chain complex of modules|chain complex]] $C_{\bullet}(X, A)$ is called the **$i$th relative homology** of the [[topological pair|pair]] $(X,A)$ and denotd $H_{i}(X,A)$.
One says relative homology is given by **relative cycles** — chains $c$ whose boundary $d_{n}(c)$ is a chain in $A$ (so that $\overline{d_{n}}(c)=[d_{n}(c)]=0$) — modulo relative boundaries — ...
$C_{n}(X,A)=\frac{C_{n}(X)}{C_{n}(A)}$ is 'chains in $X$, but identify any two chains which differ by a chain in $A