---- > [!definition] Definition. ([[homomorphism on cohomology induced by a cochain map]]) > A [[chain map|cochain map]] $f^{\bullet}$ between [[chain complex of modules|cochain complexes]] $C^{\bullet}$ and $D^{\bullet}$ induces a [[linear map|homomorphism]] on [[(co)homology of a complex|cohomology]] $f^{*}: H^{i}(C^{\bullet}) \to H^{i}(D^{\bullet})$ > by [[characterization of quotienting a group|defining]] $f^{*}([y]):=[f^{i}(y)].$ > > ```tikz > \usepackage{tikz-cd} > \usepackage{amsmath} > \begin{document} > \begin{tikzcd} > \text{ker }d^i_C \arrow[d] \arrow[r, "f^i"] & \text{ker }d^i_D \arrow[r, "\pi", two heads] & H^i(D^\bullet) \\ > H^i(C^\bullet) \arrow[rru, "\exists ! f^*"'] & & > \end{tikzcd} > \end{document} > ``` > > > Similarly, there is the notion of [[homomorphism on homology induced by a chain map]]. ---- #### ---- #### 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 > ```