---- > [!definition] Definition. ([[singular homology]]) > Let $X$ be a [[topological space]] and define $C_{n}(X)$ to be the [[free abelian group]] generated by the set of $n$-[[singular simplex|singular simplices]] in $X$. > > Elements of $C_{n}(X)$ are called **(singular) $n$-chains**; each looks like a finite formal sum $\sum_{i}n_{i} \sigma_{i}$ for $n_{i} \in \mathbb{Z}$ and $\sigma_{i}: \Delta^{n} \to X$. > > The **boundary map** $d_{n}:C_{n}(X) \to C_{n-1}(X)$ is defined as $\sigma \mapsto \sum_{i=0}^{n} (-1)^{i} \sigma \circ \delta_{i}$ > and then extending linearly. > > With this, we may define a [[chain complex of modules|chain complex]] $\dots \xrightarrow{d_{n+1}}C_{n} \xrightarrow{d_{n}}C_{n-1} \xrightarrow{d_{n-1}} \cdots C_{0} \xrightarrow{d_{0}} 0 \to \cdots$ > whose [[(co)homology of a complex|homology]] will be called the **singular homology** of $X$ and denoted $H_{\bullet}(X):=H_{\bullet}(C_{\bullet} ,d_{\bullet})$. > $H_{n}(-)$ is a [[covariant functor|covariant]] [[covariant functor|functor]] $\mathsf{Top} \to \mathsf{Ab}$ (or $\mathsf{Top} \to R\text{-}\mathsf{Mod}$ if considering [[(co)homology with coefficients|homology with coefficients]] in a general [[commutative ring|commutative]] [[ring]] $R$), obtained by composing the '[[singular (co)chain map and homomorphism induced by a continuous map|singular chain group functor]]' $C_{n}(-):\mathsf{Top} \to \mathsf{Chain}(\mathsf{Ab})$ with the $n$th [[homomorphism on homology induced by a chain map|homology functor]] $\mathsf{Chain}(\mathsf{Ab})\to \mathsf{Ab}$. > [!basicexample] > Computing singular homology groups directly is generally intractable, unless we have the very simple case of a point: > > Consider the one-point space $\text{pt}=\{ * \}$. We claim that $H^{n}(\text{pt})=H_{n}(\text{pt})=\begin{cases} > \mathbb{Z} & n = 0; \\ > 0 & n > 0. > \end{cases}$ > To see this, note that for each $n$, there is exactly one $n$-[[singular simplex]] $\sigma_{n}:\Delta^{n} \to \text{pt}$. So we have $C_{n}(\text{pt})=\mathbb{Z}\langle \sigma_{n} \rangle$. > > The boundary map $d_{n}:C_{n}(\text{pt}) \to C_{n-1}(\text{pt})$ is $d_{n}(\sigma_{n})=\sum_{i=0}^{n} (-1)^{i} \sigma_{n} \circ \delta_{i}=\sum_{i=0}^{n} (-1)^{i} \sigma_{n-1}=\begin{cases} > \sigma_{n-1} & n \text{ even};\\ > 0 & n \text{ odd}. > \end{cases}$ > So the singular [[chain complex of modules|chain complex]] looks like $\cdots \xrightarrow{0} \mathbb{Z} \xrightarrow{1} \mathbb{Z} \xrightarrow{0} \mathbb{Z} \xrightarrow{1} \mathbb{Z} \xrightarrow{0} \mathbb{Z} \to 0.$ > The claim follows. The computation for [[singular cohomology|singular]] [[(co)homology of a complex|cohomology]] is similar. ---- #### - [ ] I am sure there are more sophisticated ways to think about this [[category|category-theoretically]] [^1]:Legacy notation (matching that in [[simplicial (co)homology]]): $d_{n}(\sigma)=\sum_{i}(-1)^{i}\sigma |_{[v_{0},\dots, \widehat{v_{i}} , \dots, v_{n}]};$implicit in this definition is the canonical identification of $[v_{0},\dots,\widehat{v_{i}},\dots,v_{n}]$ with $\Delta^{n-1}$, preserving the order of vertices, so that $\sigma |_{[v_{0},\dots,\widehat{v_{i}}, \dots, v_{n}]}$ is regarded as a map $\Delta^{n-1} \to X$, that is, a singular $(n-1)$-simplex. ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```