Cohomolgy-Long exact sequence

So far the following maps are defined.

• Cochain map $\phi:H^k(A)\rightarrow H^k(B)$ induced cohomology map
  \begin{equation}
    \phi^{*}:H^k(A) \rightarrow H^k(B)
  \end{equation}
• For short exact sequence of cochain complexes \begin{equation}
    0  \rightarrow \mathcal{A} \xrightarrow{i} \mathcal{B} \xrightarrow{j} \mathcal{C} \rightarrow 0
  \end{equation}
  Connecting homomorphism map is
  \begin{equation}
    d^{*}:H^k(\mathcal{C}) \rightarrow H^{k+1}(\mathcal{A})
  \end{equation}
• Then the short exact sequence of cochain complexes
  \begin{equation}
    0  \rightarrow \mathcal{A} \xrightarrow{i} \mathcal{B} \xrightarrow{j} \mathcal{C} \rightarrow 0
  \end{equation}
  gives rise to long exact sequence in cohomology.
  \begin{equation}
   \cdots H^{k-1}(\mathcal{C}) \xrightarrow{d^{*}} H^k(\mathcal{A})  \xrightarrow{i^{*}} H^k(\mathcal{B}) \xrightarrow{j^{*}} H^k(\mathcal{C}) \xrightarrow{d^{*}} H^{k+1}(\mathcal{A}) \cdots
  \end{equation}