----- > [!proposition] Proposition. ([[chain homotopy equivalence induces isomorphism on homology]]) > If $f_{\bullet}:C_{\bullet} \to D_{\bullet}$ is a [[chain homotopy equivalence]], then the [[homomorphism on homology induced by a chain map|induced homomorphism]] $f_{*}:H_{n}(C_{\bullet}) \to H_{n}(D_{\bullet})$ is an [[isomorphism]] for all $n$. ^proposition > [!proof]+ Proof. ([[chain homotopy equivalence induces isomorphism on homology]]) > Let $g_{\bullet}$ be the chain homotopy inverse; then using functoriality and that homotopic [[chain map|chain maps]] induce the same [[linear map|homomorphism]] we have for arbitrary $n$ that $\begin{align} f_{*} \circ g_{*} = (f_{\bullet} \circ g_{\bullet})_{*} = (\id_{D_{\bullet}})_{*} = \id_{H_{n}(D_{\bullet})} \end{align}$ and vice versa. ----- #### ---- #### 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 > ```