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Given a short exact sequence of complexes, there exists a connecting homomorphism $\partial_\ast:H_k(C_\ast)\longrightarrow H_{k-1}(A_*)$, defined by

$$\partial_\ast([c])=\left[\left(f_{k-1}\right)^{-1}\left(\partial^B_k\left(\left(g_k\right)^{-1}(c)\right)\right)\right]$$


such that the complex

$$\cdots\xrightarrow{}H_k(A_*)\xrightarrow{f_k}H_k(B_*)\xrightarrow{g_k}H_k(C_*) \xrightarrow{\partial_\ast}H_{k-1}(A_*)\xrightarrow{f_{k-1}}\cdots$$


is an exact sequence

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[Cohomological version]

Given a short exact sequence of complexes, there exists a connecting homomorphism $\mathrm{d}^\ast:H^k(C^\ast)\longrightarrow H^{k+1}(A^*)$, defined by

$$\mathrm{d}^\ast([c])=\left[\left(f^{k+1}\right)^{-1}\left(\mathrm{d}_B^k\left(\left(g^k\right)^{-1}(c)\right)\right)\right]$$


such that the complex

$$\cdots\xrightarrow{}H^k(A^*)\xrightarrow{f^k}H^k(B^*)\xrightarrow{g^k}H^k(C^*) \xrightarrow{\mathrm{d}^\ast}H^{k+1}(A^*)\xrightarrow{f^{k+1}}\cdots$$


is an exact sequence

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