---- > [!definition] Definition. ([[homomorphism on homology induced by a chain map]]) > A [[chain map]] $f_{\bullet}$ between [[chain complex of modules|complexes]] $C_{\bullet}$ and $D_{\bullet}$ induces a [[linear map|homomorphism]] on [[(co)homology of a complex|homology]] $f_{*}:H_{i}(C_{\bullet}) \to H_{i}(D_{\bullet})$ by defining $f_{*}([x]) := [f_{i}(x)].$ > > > > > > > > > ```tikz > \usepackage{tikz-cd} > \usepackage{amsmath} > \begin{document} > % https://tikzcd.yichuanshen.de/#N4Igdg9gJgpgziAXAbVABwnAlgFyxMJZABgBpiBdUkANwEMAbAVxiRAB12cYAPHYANYwATgAIAvlAD6WAHoBhEONLpMufIRQBGclVqMWbTtz6CRE6XIAiSlSAzY8BIgCZd1es1aIQACRkAFFZSnABGTAwMMDgAlLaqjhpEZFp6noY+-lgB8iHs4ZHRceJ6MFAA5vBEoABmwhAAtkg6IDgQSG4gABYwdFBsOADuED19CB4G3hzsaFjxIHWNSGSt7YgAzMq19U2IK23NE15sNTLzi7vr1AeInelTnLxYcDhwogCEoqcAVCDUDHRQjAGAAFNROTQgYRYcpdHBKCjiIA > \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 cohomology induced by a cochain map]]. > [!justification] > We must show $f_{*}$ is [[well-defined]]. >- Let $[x] \in H_{n}(C_{\bullet})=\frac{Z_{n}(C_{\bullet})}{B_{n}(C_{\bullet})}$ be represented by $x \in Z_{n}(C_{\bullet})$. The element $f_{n}(x) \in D_{n}$ is a cycle, since $d_{n} \circ f_{n}(x)=f_{n-1} \circ d_{n}(x)=0$ (because $x$ is a cycle). Thus we indeed have $f_{n}(x)B_{n}(C_{\bullet}) \in H_{n}(C_{\bullet})$. >- (Invariant under changing representative) If $[x]=[y]$, then $x-y \in B_{n}(C_{\bullet})$, that is, $x-y=d_{n-1}(z)$ for some $z \in C_{n+1}$. Then $f_{n}(x)-f_{n}(y)=f_{n} \circ d_{n-1}(z)-d_{n+1} \circ f_{n+1}(z)$ is a boundary, so $[f_{n}(x)]=[f_{n}(y)]$. ^justification Alternate take: - $f_{*}$ indeed outputs elements in $H_{i}(D_{\bullet})$, because $f_{i}:C_{i} \to D_{i}$ restricts to a map $\operatorname{ker }d_{i}^{C} \to \operatorname{ker }d_{i}^{D}$. Indeed, if $x \in \operatorname{ker }d_{i}^{C}$, then $d_{i}^{D}\big( f_{i}(x) \big)=f_{i-1}\big(d_{i}^{C}(x)\big)=f_{i-1}(x)=0$. - $f_{*}$ is [[well-defined]] [[characterization of quotienting a group|because]] $\operatorname{im }d_{i+1}^{C} \subset \operatorname{ker }(\pi \circ f_{i})$ (cf. below diagram). ```tikz \usepackage{tikz-cd} \usepackage{amsmath} \begin{document} % https://tikzcd.yichuanshen.de/#N4Igdg9gJgpgziAXAbVABwnAlgFyxMJZABgBpiBdUkANwEMAbAVxiRAB12cYAPHYANYwATgAIAvlAD6WAHoBhEONLpMufIRQBGclVqMWbTtz6CRE6XIAiSlSAzY8BIgCZd1es1aIQACRkAFFZSnABGTAwMMDgAlLaqjhpEZFp6noY+-lgB8iHs4ZHRceJ6MFAA5vBEoABmwhAAtkg6IDgQSG4gABYwdFBsOADuED19CB4G3hzsaFjxIHWNSGSt7YgAzMq19U2IK23NE15sNTLzi7vr1AeInelTnLxYcDhwogCEoqcAVCDUDHRQjAGAAFNROTQgYRYcpdHBKCjiIA \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} ``` Indeed, generally $y \in \operatorname{ker }(\pi \circ f_{i})$ iff $f_{i}(y) \in \operatorname{im }d_{i+1}^{D}$. if $y \in \im d_{i+1}^{C}$, say, $y=d_{i+1}^{C}(x)$, then $f_{i}(y)=f_{i} \circ d_{i+1}^{C}(x)= d_{i+1}^{D} \circ f_{i+1}(x) \in \im d_{i+1}^{D}$. > [!basicproperties] > If $f_{\bullet}$ and $g_{\bullet}$ are [[chain homotopy|chain homotopic]] then $f_{*}=g_{*}$. Compare to the analogous property in [[homomorphism of fundamental groups induced by a continuous map]]. > - [ ] Proof is [[TODO]] (eh? definitely did it) > - [ ] any [[category]] theoretic interpretation? ^properties ---- #### ---- #### 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 > ```