-----
> [!proposition] Proposition. ([[reduced homology of a good sum of wedges is direct sum of reduced homologies]])
> For a [[wedge sum]] $\bigvee_{\alpha} X_{\alpha}$, the [[inclusion map|inclusions]] $\iota_{\alpha}:X_{\alpha} \xhookrightarrow{} \bigvee_{\alpha}X_{\alpha}$ induce an [[isomorphism]] $\bigoplus_{\alpha}\iota_{\alpha *}: \bigoplus_{\alpha} \widetilde{H}_{n}(X_{\alpha}) \to \widetilde{H}_{n}(\bigvee_{\alpha} X_{\alpha}),$
provided that the [[wedge sum]] is formed at basepoints $x_{\alpha} \in X_{\alpha}$ such that the pairs $(X_{\alpha}, x_{\alpha})$ are [[good pair|good]].
^proposition
> [!proof]+ Proof. ([[reduced homology of a good sum of wedges is direct sum of reduced homologies]])
> This is essentially a corollary of [[relative homology for a good pair is reduced homology of the quotient]].
>
The [[wedge sum]] $\bigvee_{\alpha}X_{\alpha}$ is formed as the [[quotient space|quotient]] $(\coprod_{\alpha}X_{\alpha}) / (\coprod_{\alpha}x_{\alpha})$. With $x_{0}=\coprod_{\alpha}x_{\alpha} / \coprod_{\alpha}x_{\alpha}$ the wedge point of it, we have $\begin{align}
\widetilde{H} _{n}(\bigvee_{\alpha}X_{\alpha}) & = H_{n}(\bigvee_{\alpha}X_{\alpha}, x_{0}) \\
& = H_{n} \big(\coprod_{\alpha}X_{\alpha} / \coprod_{\alpha}x_{\alpha} , \coprod_{\alpha}x_{\alpha} / \coprod_{\alpha}x_{\alpha} \big) \\
&= H_{n}(\coprod_{\alpha}X_{\alpha} , \coprod_{\alpha}x_{\alpha}) \\
& = \bigoplus_{\alpha} H_{n}(X_{\alpha}, x_{\alpha}) \\
& = \bigoplus \widetilde{H}_{n}(X_{\alpha}),
\end{align}$
where we have used [[homology of disjoint union is direct sum of homologies]].
^4af639
-----
####
----
#### 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
> ```