---- > [!definition] Definition. ([[reduced homology]]) > If $(X, x_{0})$ is a [[pointed set|pointed]] [[topological space]], the **reduced homology** of $X$ is its [[singular homology|homology]] [[relative singular homology|relative to]] the basepoint: $\widetilde{H}_{n}(X):=H_{n}(X, \{ x_{0} \}).$ > This is explicitly given by $\widetilde{H}_{n}(X) \xrightarrow{\cong}\begin{cases} H_{n}(X) & n \geq 1; \\ H_{0}(X) / \mathbb{Z}\langle x_{0} \rangle & n = 0. \end{cases}$ > [!justification] > **If $n\geq 2$:** Using the [[long exact sequence for relative singular homology]], have an [[exact sequence]] $\cdots \to \cancel{ H_{n}(x_{0}) }^{0} \xrightarrow{\iota_{*}} H_{n}(X) \xrightarrow{q_{*}} \widetilde{H}_{n}(X) \xrightarrow{\partial_{}} \cancel{ H_{n-1}(x_{0}) }^{0} \to \cdots$ > implying $q_{*}$ is an [[isomorphism]]. > > **If $n=1$.** (Not sure why we skipped this, come back) > > **If $n=0$:** have an [[exact sequence]] > > $\begin{align} > \cdots \to \widetilde{H}_{1}(X) \to & \underbrace{ H_{0}(\{ x_{0} \}) }_{ \cong \mathbb{Z} } \xrightarrow{\iota_{*}}H_{0}(X) \twoheadrightarrow \widetilde{H}_{0}(X) \to 0. \\ > 1 & \ \ \ \ \ \mapsto \ \ \ \ \ \ \ \ \ \ \ \ \ \ [x_{0}] > \end{align}$ > > $\widetilde{H}_{0}(X)$ is the [[cokernel of a module homomorphism|cokernel]] of $\iota_{*}$ in this sequence, which is $H_{0}(X) / \mathbb{Z} \langle x_{0} \rangle$. ---- #### ---- #### 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 > ```